department of mathematics, faculty of mathematical...

Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

The 7th Seminar on Harmonic Analysis and Applications(January 17-18, 2019)

and Workshop on Dynamical Systems(January 19-20, 2019)

Department of Mathematics,

Faculty of Mathematical Sciences,

Shahid Beheshti University

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Faculty of Mathematical Sciences, Shahid Beheshti University

The faculty of Mathematical Sciences was established in 1968 and consisted of two departments ofstatistics and mathematics. Later in 1999 the department of computer sciences and in 2017 the departmentof applied and industrial mathematics were established. The faculty has organized several conferences pastfew years, namely• 24th Iranian Mathematics Conference (IMC) in April 1994• 24th International Statistics Conference in August 1998• 1st Students Seminar on Statistics: August 1999• 12th Seminar of Algebra in April 2000• 1st Seminar of Insurance Statistics in August 2000• 1st Seminar on philosophy of Mathematics, Oct 2001• 1st Workshop on the History of Mathematics, Oct 2004• Stochastic Differential Equations of Mathematical Finance June 18-26 2005• The 1st Workshop on Applied Mathematical joint between Iran and France• Annual Seminar of Blossoms of Math. From 1996 to 2000• Center of Excellence in Algebraic and Logical structures in Discrete Mathematics and their ApplicationsDepartment of Mathematics commenced presenting PhD in different fields of functional analysis, algebraicstructures, and dynamic systems in 1994. At the time being, different courses in different levels are available.The complete list is presented below.

Department Members

No. Name Academic Rank1 Mohammad Mahdi Ebrahimi Professor2 Rajab Ali Borzoei Professor3 Mahdi Pourbarat Abozed Abadi Assistant Professor4 Samad Hajjabbari Assistant Professor5 Alireza Hosseiniuoon Professor6 Abbas Fakhari Associate Professor7 Mojgan Mahmoudi Professor8 Morteza Moniri Associate Professor9 Vida Millani Assistant Professor10 Khosro Monsef Shokri Assistant Professor11 Alireza Salemkar Professor12 Negar Shahani Karamzadeh Assistant Professor13 Masood Toosi Ardakani Professor14 Reza Taleb Assistant Professor

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Founding Fellows:

Seyyed Masoud Amini (Tarbiat Modares University, Iran)

Mohammad Ali Dehghan (Vali-e-Asr University of Rafsanjan, Iran)

Hamid Reza Ebrahimi Vishki (Ferdowsi University of Mashhad, Iran)

Gholam Hossein Eslamzadeh (University of Shiraz, Iran)

Taher Ghasemi Honari (Kharazmi University, Iran)

Seyed Ali Reza Hosseinion (Shahid Beheshti University, Iran)

Rajab Ali Kamyabi Gol (Ferdowsi University of Mashhad, Iran)

Mahmood Lashkarizadeh Bami (University of Isfahan, Iran)

Alireza Medghalchi (Kharazmi University, Iran)

Rasoul Nasr Isfahani (Isfahan University of Technology, Iran)

Abdolrasoul Pourabbas (Amirkabir University of Technology, Iran)

Mohammad Ali Pourabdollah Nejad (Ferdowsi University of Mashhad, Iran)

Mehdi Rajabalipour (Shahid Bahonar University of Kerman, Iran)

Ali Rejali (University of Isfahan, Iran)

Abdolhamid Riazi (Amirkabir University of Technology, Iran)

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Scientific Committee:

Mohammad Akbari Tootkaboni (University of Guilan, Iran)

Seyyed Masoud Amini (Tarbiat Modares University, Iran)

AliAkbar Arefi Jamal (University of Sabzevar, Iran)

Ataollah Askari Hemmat (Shahid Bahonar University of Kerman, Iran)

Hamid Reza Ebrahimi Vishki (Ferdowsi University of Mashhad, Iran)

Abbas Fakhari (Local organizer) (Shahid Beheshti University, Iran)

Rajab Ali Kamyabi Gol (Ferdowsi University of Mashhad, Iran)

Alireza Medghalchi (Kharazmi University, Iran)

Rasoul Nasr Isfahani (Isfahan University of Technology, Iran)

Meysam Nassiri (IPM, Iran)

Abdolrasoul Pourabbas (Amirkabir University of Technology, Iran)

Hamidreza Rahimi ( Islamic Azad University, Tehran North Branch, Iran)

Mohammad Ramezanpour (University of Damghan, Iran)

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Executive Committee:

Bijan Ahmadi (Shahid Beheshti University, Iran)

Rajab Ali Borzouie (Shahid Beheshti University, Iran)

Abbas Fakhari (Local organizer) (Shahid Beheshti University, Iran)

Mojtaba Ganjali (Shahid Beheshti University, Iran)

Masoud Hajarian (Shahid Beheshti University, Iran)

Fatemeh Khosravi (IPM, Iran)

Mojhgan Mahmoodi (Shahid Beheshti University, Iran)

Morteza Moniri (Shahid Beheshti University, Iran)

Khosro Monsef Shokri (Shahid Beheshti University, Iran)

Mahdi Pourbarat (Shahid Beheshti University, Iran)

Alireza Salemkar (Shahid Beheshti University, Iran)

Mohammad Sadegh Shahrokhi Dehkordi (Shahid Beheshti University, Iran)

Reza Taleb (Shahid Beheshti University, Iran)

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Invited Speakers:

Jose Alves (Porto University, Portugal)

Mohammad Ali Dehghan (Vali-e-Asr University of Rafsanjan, Iran)

Peyman Eslami (Warwick University, UK)

Gholamhossein Eslamzadeh (University of Shiraz, Iran)

Mahya Ghandehari (University of Delaware, USA)

Mehrdad Kalantar (University of Houston, USA)

Adam Skalski (IMPAN, Poland)

Mohammad Soufi (Universidade Federal do Rio de Janeiro, Brazil)

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Schedule of

The 7th Seminar on Harmonic Analysisand Applications

Thursday Jan. 17

8-8:30 Registration

8:30-9:30 Opening

9:30-10:15 Speech: Dr. Mohammad Ali Dehghan

Frames and some of their topologies and collections

10:15-10:45 Coffee Break & Conference photos

10:45-11:10

Short

Talks

Ramezanpoor: Cyclic amenability of Lau product of Banach algebras

Rashidi: Duality of controlled frames

Arabiani: Optimal dual frames in Hilbert spaces

شمس: توابع ناوردا با هسته های ماتریس مقدار

11:15-11:40

Short

Talks

Samadzadeh: Some properties of woven frames and their related operators

Gandomani: On generalized right derivations of Banach algebras related to

locally compact abelian groups

Shamsabadi: O-cross Gram matrices with respect to g-frames

Yousefiazar: Orlicz algebra Lφ(S)

11:45-12:10

Short

Talks

Moazzami: On convergence problem of Fourier series

Lakzian: Topological center of Generalized Matrix Banach Algebras

Momeni: A note on module generalized derivations of triangular Banach algebras

کاربرد عمل گروه برای یافتن معادله برآورد در یک مدل با پارامتر مزاحم شمس:

12:10-12:30

Posters

Alishahi: Generalization of the Hausdorff-Young inequality for Pseudo-integral

دوگانی کانونی قاب های موجکحقیقت جو:

Pirali: Some conditions of derivations of tensor products on nonassociative

algebras and some results about generalized derivations on C*-algebras

فشرده-G توپولوژیک راست-G برخی خواص جبری روی نیم گروههایبایمانی:

Shabani: Specification property for the iterated function systems

12:30-14 Lunch

14-14:30

Short

Talks

Shirinkalam: A Banach algebra associated with a locally compact groupoid

Razghandi: On the representation frames and duality

Yousefiazar: Multipliers on A1(G/H) and Lp(G/H)

Hassani: On the uniform distribution of sequences involving the imaginary parts

of the non-trivial zeros of the Riemann zeta function

14:30-15:15 Speech: Dr. Mehrdad Kalantar

A notion of topological boundary for unitary representations

15:15-15:45 Coffee Break

15:45-16:10

Short

Talks

Amin. Khosravi: The second dual of a Jordan triple system

Ghasemi: Centralizing Derivations on Banach Algebras Related to Locally

Compact Groups

Rashidi: Some notes on controlled frames in Hilbert C*-modules

Zaj: Banach space valued mappings with compact image

Friday Jan. 18

8-8:25

Short

Talks

Shams Yousefi: On the Fourier algebra of a C*-dynamical system

Sadeghi: Jordan amenability of Lau product of Jordan Banach algebras

Enayati: Characterizing g-R-duality in Hilbert spaces

Basati: Woven frames in Banach spaces

8:30-8:55

Short

Talks

Alinejad: Quasi-multipliers on Banach algebras related to locally compact

groups

Shariati: Module Johnson amenability of Banach algebras

Movahed: Characterization of p-frames in light of Hahn-Banach Theorem

Rahimi: Perturbation of woven-weaving fusion frames in Hilbert spaces

9-9:45 Speech: Dr. Gholam Hossein Eslamzadeh

Existence of projections properties in C*-algebras


10:15-11 Speech: Dr. Mahya Ghandehari

On non-commutative weighted Fourier algebras

11:05-11:30

Short

Talks

Askari-sayah: Johnson pseudo-contractibility of certain Banach algebras and

their nilpotent ideals

Feyzi: A note on Bochner-Schenberg-Eberlin algebras

Basati: Some results on woven frames in Banach Spaces

Ghobadzadeh: Properties of g-frame representations with bounded operators

11:35-12

Short

Talks

Alimohammadi: Arens regularity of real Banach algebras and (-1)-weak

amenability of second dual of real Banach algebras

Lazkian: Best proximity points for weak MT-cyclic Reich

Dashti: Vector-valued φ-contractibility of Banach algebras

Fatemeh Khosravi: Kawada-Itô theorem for locally compactquantum groups

12-12:30 Closing

12:30-13 Lunch

Workshop on Dynamical Systems

January 19-20, 2019

Sun. Jan. 20 9-10:15 Alikhan


10:45-12 Jose Alves

12-13:30 Lunch

13:30-14:45 Eslami


15:15-16:30 Discussion

Sat. Jan. 19 9-10:15 Soufi


10:45-12 Jose Alves

12-13:30 Lunch

13:30-14:45 Eslami


15:15-16:30 Skalski

Abstracts of the Invited Speakers(In Alphabetical Order)

Abstracts of the Invited Speakers

Frames and some of their topologies andcollections

Mohammad Ali Dehghan(Vali-e-Asr University of Rafsanjan, Iran)

Abstract: Frames as a subset of Bessel sequences have some topological structures. Riesz Bases,overcomplete and tight frames make interesting open and closed sets in these topologies. We are goingto find some subcollections of Bessel sequences or frames and Riesz bases that have a particular property.These collections have an special structure that give us suitable frames with potential applications in wirelesssensor network and preprocessing of signals. This is based on a joint work with Akram Bibak and MehdiMesbah.

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Existence of projections properties in C∗-algebras

Gholam Hossein Eslamzadeh(Shiraz University, Shiraz, Iran)

Abstract: We characterize the small projections property (SP) for type I C∗-algebras with Hausdorffprimitive spectrum. As an application to group C∗-algebras, we identify certain locally compact groupsG for which C∗(G) has (SP). Finally, among other results, a characterization of AW ∗-algebras is given interms of (SP). This talk is based on joint work with Milad Moazami Goodarzi

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On non-commutative weighted Fourier algebras

Mahya Ghandehari(University of Delaware, USA)

Abstract: Beurling-Fourier algebras are analogues of the Beurling algebra in the non-commutativesetting. These algebras for general locally compact groups were defined by Lee and Samei as the predualof certain weighted von Neumann algebras, where a weight on G is defined to be a suitable unboundedoperator affiliated with the group von Neumann algebra. In this talk, we present the general definition ofa Beurling-Fourier algebra, and discuss how their spectra can be identified. In particular, we determinethe Gelfand spectrum of Beurling-Fourier algebras for some representative examples of Lie groups, such asSU(n), the Heisenberg group, and the Euclidean motion group, emphasizing the connection of spectra tothe complexification of underlying Lie groups. This talk is based on joint work with Lee, Ludwig, Spronk,and Turowska.

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A notion of topological boundary for unitaryrepresentations

Mehrdad Kalantar(University of Houston, USA)

Abstract: Abstract: We introduce a generalization of the notion of the Furstenberg boundary of adiscrete group G to the setting of a general unitary representation π : G → B(Hπ). This space, whichwe call the “Furstenberg-Hamana boundary” is a G-invariant subspace of B(Hπ) that carries a canonicalC∗-algebra structure. In many natural cases, including when π is a quasi-regular representation or, moregenerally, a Koopman representation, the Furstenberg-Hamana boundary of π is commutative, hence of theform C(X) for a compact G-space X. We give a dynamical characterization of this boundary in the case ofquasi-regular representations. We give few examples and applications. This is joint work with Alex Bearden.

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Quantum Dirichlet forms and their applicationsto locally compact quantum groups

Adam Skalski(IMPAN, Poland)

Abstract: Theory of Dirichlet forms is a crucial tool in the analysis of classical Markov semigroups.Following the work of Goldstein, Lindsay, and Cipriani we will first outline general results on quantumDirichlet forms in the most general context of von Neumann algebras equipped with weights. Then we willshow how in presence of translation invariance these can be used in the study of locally compact quantumgroups and the related convolution semigroups of states. Some examples, applications to approximationproperties and very recent results on generating functionals for convolution semigroups will be also presented.Based on joint work with Ami Viselter.

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Extended Abstracts of the ContributedTalks

(In Alphabetical Order)

On generalized right derivations of Banach algebras related to locallycompact abelian groups

M. H. Ahmadi Gandomani* (Ghirokarzin Branch, Islamic Azad University, Fars, Iran)M. J. Mehdipour (Shiraz University of Technology, Shiraz, Iran)

abstract: Let G be a locally compact abelian group and L∞0 (G) be the Banach space of all essentiallybounded measurable functions on G vanishing at infinity. In this paper, we present some results concern-ing generalized right derivations on the noncommutative Banach algebra L∞0 (G)∗. We prove that everygeneralized right derivation D on L∞0 (G)∗ maps L1(G) into L1(G). Moreover D maps rad(L∞0 (G)∗) =Annr(L

∞0 (G)∗) into (0). We also investigate centralizing generalized right derivations of L∞0 (G)∗ and prove

that D is centralizing if and only if D is a right centralizer.Finally, we show that there is no nonzero skew centralizing generalized right derivation on L∞0 (G)∗.

keywords. : Locally compact abelian group, right derivation, generalized right derivation, skew left cen-tralizers, centralizing mapping.

subject. 43A15, 47B47, 16W25

1 introduction

Let G be a locally compact abelian group with a fixed left Haar measure λ. The Banach space of complex-valued integrable functions with respect to λ is denoted by L1(G). With the norm ‖.‖1 and with convolution

φ ∗ ψ(x) =

∫

Gφ(y)ψ(y−1x) dλ(y) (x ∈ G)

as product, L1(G) becomes a Banach algebra. Let L∞0 (G) be the subspace of L∞(G), the usual Lebesguespace as defined in [4] equipped with the essential supremum norm ‖.‖∞, consisting of all functions g ∈L∞(G) that vanish at infinity; i.e. for each ε > 0, there is a compact subset K of G for which ‖g χG\K‖∞ < ε,where χG\K denotes the characteristic function of G \K on G.

For each φ ∈ L1(G), we may consider φ as a linear functional in the dual of L∞0 (G), represented byL∞0 (G)∗, defined by

〈φ, g〉 =

∫

Gg(x)φ(x) dλ(x) (g ∈ L∞0 (G)).

It is well-known from [5] that L∞0 (G)∗ is a Banach algebra with the first Arens product “ · ” defined by theformula

〈m · n, g〉 = 〈m,ng〉, ,

19

M. H. Ahmadi Gandomani, M. J. Mehdipour

where the functional ng is defined by

〈ng, φ〉 = 〈n, gφ〉, in which 〈gφ, ψ〉 = 〈g, φ ∗ ψ〉for all m,n ∈ L∞0 (G)∗, g ∈ L∞0 (G) and φ, ψ ∈ L1(G).

Denote by Λ(G) the set of all weak∗-cluster points of the canonical images of the bounded approximateidentities, bounded by one, of L1(G) in L∞0 (G)∗. It is easy to see that m · u = m and u · φ = φ for allm ∈ L∞0 (G)∗, φ ∈ L1(G) and u ∈ Λ(G).

The right annihilator of L∞0 (G)∗ is denoted by Annr(L∞0 (G)∗) and is defined by

Annr(L∞0 (G)∗) = r ∈ L∞0 (G)∗ : L∞0 (G)∗ · r = 0.

Let us remark from [6] thatAnnr(L

∞0 (G)∗) = rad(L∞0 (G)∗),

where rad(L∞0 (G)∗) stands for the Jacobson radical of L∞0 (G)∗.An additive mapping D on L∞0 (G)∗ is called a generalized right derivation if there exists an additive

mapping d on L∞0 (G)∗ such that

d(m · n) = d(m) · n+ d(n) ·m (1)

andD(m · n) = D(n) ·m+ d(m) · n

for all m,n ∈ L∞0 (G)∗. An additive mapping d satisfying (1) is called a right derivation. A generalized rightderivation with associated right derivation d is denoted by (D, d). Obviously generalized right derivations onL∞0 (G)∗ include right derivations and skew left centralizers, an additive mapping T on L∞0 (G)∗ satisfying

T (m · n) = T (n) ·m.let us recall that a mapping T : L∞0 (G)∗ → L∞0 (G)∗ is called centralizing if

[T (m),m] ∈ Z(L∞0 (G)∗)

for all m ∈ L∞0 (G)∗, where Z(L∞0 (G)∗) is the center of L∞0 (G)∗, the set of all m ∈ L∞0 (G)∗ such thatm · n = n ·m for all n ∈ L∞0 (G)∗. In a special case when

[T (m),m] = 0

for all m ∈ L∞0 (G)∗, T is called commuting, where

[m,n] = m · n− n ·m.

IfT (m) ·m+m · T (m) ∈ Z(L∞0 (G)∗)

for all m ∈ L∞0 (G)∗, then T is called skew-centralizing, and similarly T is called skew-commuting if

T (m) ·m+m · T (m) = 0,

for all m ∈ L∞0 (G)∗.For a non-discrete locally compact group G, an easy application of the Hahn-Banach’s theorem shows

that L∞0 (G)∗ has a non-zero element r ∈ Annr(L∞0 (G)∗). Hence

r · L∞0 (G)∗ · r = 0.

This fact implies that L∞0 (G)∗ is not a prime ring. Therefore, we cannot apply the well-known resultsconcerning derivations of prime rings to L∞0 (G)∗. It is natural to ask whether the results hold for L∞0 (G)∗.In this paper, we investigate these questions for generalized right derivations of L∞0 (G)∗.

In this paper we investigate generalized right derivations of L∞0 (G)∗ and prove that any generalized rightderivation D on L∞0 (G)∗ maps L1(G) into L1(G). Moreover D maps rad(L∞0 (G)∗) = Annr(L

∞0 (G)∗) into

(0). We also investigate centralizing generalized right derivations of L∞0 (G)∗ and prove that D is centralizingif and only if D is a right centralizer. Finally, we show that there is no nonzero skew centralizing generalizedright derivation on L∞0 (G)∗.

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On generalized right derivations of Banach algebras related to locally compact abelian groups

2 Centralizing and skew centralizing generalized right derivations

We use the following theorem from [2] (theorem 3.2).theorem Let d be a right derivation on L∞0 (G)∗. Then the following statements hold.

(i) d maps L∞0 (G)∗ into its radical.(ii) d|rad(L∞0 (G)∗) = 0.(iii) d is nilpotent.

lemma Let (D, d) be a generalized right derivation on L∞0 (G)∗. Then the following assertions hold.(i) D = LD(u) + d.(ii) D maps Annr(L

∞0 (G)∗) into (0).

(iii) D maps L1(G) into L1(G) and D(φ) = D(u) · φ for all φ ∈ L1(G).

The next theorem characterize centralizing and commuting generalized right derivations.theorem Let (D, d) be a generalized right derivation on L∞0 (G)∗. Then the following assertions are

equivalent.(a) D is commuting;

(b) D is centralizing;(c) [[D(m),m],m] = 0, for all m ∈ L∞0 (G)∗;(d) D is a right centralizer. Moreover d = 0 and D(u) ∈ L1(G), for all u ∈ Λ(G).

We finish the paper with the following result which characterizes skew centralizing generalized rightderivations of L∞0 (G)∗.

theorem Let (D, d) be a skew centralizing generalized right derivation of L∞0 (G)∗. Then D = 0.

References

[1] M. H. Ahmadi Gandomani and M. J. Mehdipour, Generalized derivations on some convolution algebras,Aequationes Math., 92 (2018), pp. 223-241.

[2] M. H. Ahmadi Gandomani and M. J. Mehdipour, Jordan, Jordan right and Jordan left derivations onconvolution algebras, Bull. Iranian Math. Soc., DOI: 10.1007/s41980-018-0125-7.

[3] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer,Berlin-Heidelberg-New York, 1973.

[4] E. Hewitt and K. Ross, Abstract Harmonic Analysis I, Springer-Verlag, New York, 1970.

[5] A. T. Lau and J. Pym, Concerning the second dual of the group algebra of a locally compact group, J.London Math. Soc., 41 (1990), pp. 445–460.

[6] M. J. Mehdipour and Z. Saeedi, Derivations on group algebras of a locally compact abelian group,Monatsh. Math., 180 (3) (2016), pp. 595-605.

[7] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), pp. 1093-1100.

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Quasi-multipliers on Banach algebras related to locally compact groups

A. Alinejad* (College of Farabi, University of Tehran, Tehran, Iran)M. Rostami (Amirkabir University of Technology, Tehran, Iran)

Abstract: In this paper, we give some characterizations for QM(L1(G)), the quasi-multipliers of L1(G)for a locally compact group G. Also we deal with the quasi-multipliers on the dual Banach algebra L∞0 (G)and prove that its qusi-multipliers is isomorphic with M(G). furthermore we describe some classes of quasi-multipliers on L∞0 (G)∗ is equivalent to discreteness of G.

keywords. Quasi-multiplier, Group algebra, Locally compact group

subject. 43A15, 43A22, 43A10

1 Introduction

A quasi-multiplier is a generalization of the notion of a multiplier. The first systematic account of thegeneral theory of quasi-multiplier on a Banach algebra with a bounded approximate identity was given in apaper by McKennon [11] in 1977. He has expressed the definition of a quasi-multiplier for a complex Banachalgebra A as follows: A bilinear mapping m : A×A −→ A such that

m(ab, cd) = am(b, c)d for all a, b, c, d ∈ A.

Let QM(A) denote the set of all separately continuous quasi-multipliers on A.When A admits a bounded ap-proximate identity, it is shown that QM(A) is a Banach space with the norm ‖m‖ = sup‖m(a, b)‖; a, b ∈A, ‖a‖ = ‖b‖ = 1, [11, Theorem 2]. For some classical Banach algebras, the Banach space of quasi-multipliers may be identified with some other known space or algebras. For example, when G is a locallycompact Hausdorff group, QM(L1(G)) is identified with the measure algebra M(G). Subsequent devel-opments have been made as a result of contributions by Vasudevan and Goel [14]. They have shown anembedding of QM(A) in the second conjugate space A∗∗ of a Banach algebra A, which extend the well-knownembedding of the left multipliers (right multipliers) of A in A∗∗.

After that, the theory of quasi-multipliers on Banach algebras was studied further by Kassem andRowlands [6], Vasudevan and Goel and Takahasi [15], Grosser [3], Argun and Rowlands [1]. See also[7, 8, 10] for improvements of quasi-multipliers on C∗-algebras and operator spaces.

Let G be a locally compact group, and L∞(G) be the usual Lebesgue space as defined in [4] equippedwith the essential supremum norm ‖ · ‖∞. The space L∞(G) with the pointwise operations, the complexconjugation as involution and the norm ‖ · ‖∞ is a commutative C∗-algebra with identity element 1.

Note that a function f ∈ L∞(G) vanishes at infinity if for each ε > 0, there is a compact subset K of Gfor which

‖fχG\K‖∞ ≤ ε;where χG\K denotes the characteristic function of G\K on G. We denote by L∞0 (G) the set of all functionsin L∞(G) that vanish at infinity; L∞0 (G) is left introverted in L∞(G). For an extensive study of L∞0 (G) seeLau and Pym [9].

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Let L1(G) be the group algebra of G defined as in [4] equipped with the convolution product ∗ and thenorm ‖ · ‖1. Remark that L∞(G) is the continuous dual of L1(G) under the usual duality. It is well knownthat L1(G) is a two-sided closed ideal in L∞0 (G)∗; see [9, Theorem 2.11]. Furthermore, an easy applicationof the Goldstine’s theorem shows that L1(G) is weak∗ dense in L∞0 (G)∗. Multipliers and compact leftmultipliers of these certain class of Banach algebras are investigated in [12, 13].

In this paper, we develope the theory of quasi-multipliers on Banach algebras related to locally compactgroups. As the main result, we prove that QM(L∞0 (G)∗) may be identified by M(G). Our next step is togive a characterization for the existence of a quasi-multiplier on the group algebra L1(G). furthermore wedescribe some classes of quasi-multipliers on L∞0 (G)∗ is equivalent to discreteness of G.

2 Main results

We commence this section with the following result which characterize the existence the quasi-multiplier onthe group algebra L1(G).

theorem 2.1. LetG be a locally compact group. For any continuous bilinear mapping m : L1(G)×L1(G) −→L1(G), the folowing statement are equivalent:

(i) m(f ′ ∗ f, g ∗ g′) = f ′ ∗m(f, g) ∗ g′, for all f, f ′, g, g′ ∈ L1(G).

(ii) m(δx ∗ f, g ∗ δy) = δx ∗m(f, g) ∗ δy, for all f, g ∈ L1(G) and x, y ∈ G.

Therefore, we have shown the following:

corollary 2.2. Let G be a locally compact group. A continuous bilinear mapping m : L1(G) × L1(G) −→L1(G), is quasi-multiplier on L1(G) if and only if, m(δx ∗ f, g ∗ δy) = δx ∗ m(f, g) ∗ δy, for all f, g ∈ L1(G)and x, y ∈ G.

theorem 2.3. Let G be a locally compact group, and let continuous bilinear mapping m : L1(G)×L1(G) −→L1(G), satisfying m(δx ∗ f, g ∗ δy) = δx ∗ m(f, g) ∗ δy, for each f, g ∈ L1(G) and x, y ∈ G. Then there isξ ∈M(G) such that m(f, g) = f ∗ ξ ∗ g. Moreover ‖m‖ = ‖ξ‖.

Let A be a Banach algebra and write LM(A) and RM(A) for the Banach algebra of left and rightmultipliers, respectively, on A. Define λ : LM(A) −→ QM(A) by, for each T ∈ LM(A), letting [λ(T )](x, y) =xT (y) for all x, y ∈ A. The map λ is a linear isometry. The function ρ : RM(A) −→ QM(A), defined by[ρ(T )](x, y) = T (x)y for all x, y ∈ A, is also a linear isometry [11, Theorem 4]. It’s always interesting toknow when the maps λ and ρ are surjective?

The restriction P(m) of m ∈ L∞0 (G)∗ of the subspace C0(G) of L∞0 (G) determines a quotient mappingP : L∞0 (G)∗ −→ C0(G)∗ = M(G). As shown in [9], for m,n ∈ L∞0 (G)∗ the product m · n depends only onP(n).

Let us recall that en element u ∈ L∞0 (G)∗ is called a mixed identity if

f · u = u · f = f (f ∈ L1(G)).

Let G be alocally compact group, we denote by E(G), the nonempty set of all mixed identities of L∞0 (G)∗,and E1(G), consisting of those with norm 1. Note that u ∈ E1(G) if and only if it is a weak∗-cluster point of anapproximate identity in Ma(S) bounded by one. Moreover, when S be a compactly cancellative foundationsemigroup with identity e, and let u be an element of L∞0 (G)∗ with norm 1, then u is a mixed identity ifand only if u is a right identity for L∞0 (G)∗ and P(u) = δe; that is,

m · u = m

for all m ∈ L∞0 (G)∗; see [9].

theorem 2.4. Let G be alocally compact group. Then the quasi-multiplier of L∞0 (G)∗ is isomorphic withM(G).

23

A. Alinejad, M. Rostami

The right annihilator of L∞0 (G)∗ is denoted by ran(L∞0 (G)∗) and is defined by

ran(L∞0 (G)∗) = r ∈ L∞0 G)∗| L∞0 (G)∗ · r = 0.

Let G be alocally compact group. Then ran(L∞0 (G)∗) is the weak∗ closed ideal

ker(P) = n− u · n| n ∈ L∞0 G)∗

in L∞0 (G)∗ for all u ∈ E1(S); see [5, P. 139].

theorem 2.5. Let G be a locally compact group and m : L∞0 (G)∗ × L∞0 (G)∗ −→ L∞0 (G)∗ be a quasi-multiplier. Then the following statements hold:

(i) m(L1(G)× L1(G)) ⊆ L1(G).

(ii) m(ran(L∞0 (G)∗)× ran(L∞0 (G)∗)) ⊆ ran(L∞0 (G)∗).

For a Banach algebra A define φ : A −→ QM(A) by, for each a ∈ A, letting φa(x, y) = xay for allx, y ∈ A. As pointed out in [11, Theorem 3], if A has a boubded approximate identity, then the map φ is alinear isometry of A into QM(A).

theorem 2.6. Let G be alocally compact group. Then any quasi-multiplier on L∞0 (G)∗ is of the form φffor some f ∈ L1(G) if and only if G is discrete.

References

[1] Z. Argun and K. Rowlands, On quasi-multipliers, studia Math. 108 (1994), 217–245.

[2] B. Dearden, Quasi-multipliers of Pedersen’s ideal, Rocky Mountain J. Math. 22 (1992), 157–163.

[3] M. Grosser, Quasi-multipliers of the algebra of approximable operators and its duals, Studia Math. 124(1997), 291–300.

[4] E. Hewitt and K. Ross, Abstract Harmonic Analysis I, (Springer, New York, 1970).

[5] N. Isik, J. Pym and A. ulger, The second dual of the group algebra of a compact group, J. London Math.Soc. 35 (1987), 135-148.

[6] M. S. Kassem and K. Rowlands, The quasi-stric topology on the space of quasi-multipliers of aB∗−algebras, Math. Proc. Cambridge Philos. Soc. 101 (1987), 555–566.

[7] M. Kaneda, Quasi-multipliers and algebrizations of an operator space, J. Funct. Anal. 251 (2007), 347–365.

[8] M. Kaneda and V. I. Paulsen, Quasi-multipliers of operator spaces, J. Funct. Anal. 217 (2004), 346–359.

[9] A. T. Lau and J. Pym, Concerning the second dual of the group algebra of a locally compact group, J.London Math. Soc. 41 (1990), 445-460.

[10] H. Lin, Support algebras of σ−unital C∗−algebras and their quasi-multipliers, Trans. Amer. Math. Soc.325 (1991), 829–854.

[11] Kelly McKennon, Quasi-multipliers, Trans. Amer. Math. Soc. 233, (1977), 105-123.

[12] M. J. Mehdipour and R. Nasr-Isfahani, Completely continuous elemnts of Banach algebra related tolocally compact groups, Bull. Aust. Math. Soc. 76 (2007), 49–54.

[13] M. J. Mehdipour and R. Nasr-Isfahani, Compact left multipliers on a Banach algebra related to locallycompact groups, Bull. Aust. Math. Soc. 79 (2009), 227–238.

24


[14] R. Vasudevan and Satya Goel, Embedding of Quasi-multipliers of a Banach algebra in its second dual,Math. Proc. Camb. Phil. Soc. 95 (1984), 457–466.

[15] R. Vasudevan, S. Goel and S. Takahasi, The Arens product and quasi-multipliers, Yokohama. Math. J.33 (1985), 49–66.

25

Arens regularity of real Banach algebras and (−1)-weak amenability ofsecond dual of real Banach algebras

D. Alimohammadi* and H. Alihoseini

(University of Arak, Arak, Iran)

Abstract: Let (A, ‖ · ‖) be a real Banach algebra, a complex algebra AC be a complexification of Aand ‖| · ‖| be an algebra norm on AC satisfying a simple condition together with the norm ‖ · ‖ on A. Wefirst show that A is Arens regular if and only if AC is Arens regular. Next we prove that A∗ is a realBanach A∗∗-module if and only if (AC)∗ is a complex Banach (AC)∗∗-module. Finally, we show that A∗∗ is(−1)-weakly amenable if and only if (AC)∗∗ is (−1)-weakly amenable.

keywords. Banach algebra, Banach module, Complexification, Derivation, (−1)-Weak amenability.

subject. 46H25, 46H20

1 Introduction And Preliminaries

The symbol F denotes a field that can be either R or C. For a Banach space X over F we denote by X∗ andX∗∗ the dual space and the second dual space of X, respectively.

Let B be an algebra and X be a B-module over F with the module operations (a, x) 7−→ a · x, (a, x) 7−→x · a : B × X −→ X. A linear map D : B −→ X over F is called an X-derivation on B over F if D(ab) =D(a)·b+a·D(b) for all a, b ∈ B. For each x ∈ X, the map δx : B −→ X defined by δx(a) = a·x−x·a (a ∈ B),is an X-derivation on B over F. An X-derivation D on B is called inner if D = δx for some x ∈ X.

Let (B, ‖ · ‖) be a Banach algebra over F. A B-module X over F is called a Banach B-module if X is aBanach space with a norm ‖ · ‖ and there exists a positive constant k such that

‖a · x‖ ≤ k‖a‖‖x‖, ‖x · a‖ ≤ k‖a‖‖x‖,

for all a ∈ B and x ∈ X. Clearly, B is a Banach B-module over F with the module operations a · b = ab andb · a = ba for all a, b ∈ B. Let X be a Banach B-module over F with the module operations (a, x) 7→ a · x,(a, x) 7→ x · a : B × X −→ X. Then X∗ is a Banach B-module over F with the natural module operations(λ, a) 7−→ a · λ, (λ, a) 7−→ λ · a : B × X∗ −→ X∗ given by

(a · λ)(x) = λ(x · a), (λ · a)(x) = λ(a · x) (a ∈ B, λ ∈ X∗, x ∈ X),

and with the operator norm ‖ · ‖op. In particular, B∗ is a Banach B-module over F. We denote by Z1F(B,X)

the set of all continuous X-derivations on B over F. Clearly, Z1F(B,X) is a linear space over F which contains

all inner X-derivations on B over F. We denote by N1F(B,X) the set of all inner X-derivations on B over

F. Clearly, N1F(B,X) is a linear subspace of Z1

F(B,X) over F. We denote by H1F(B,X) the quotient space

Z1F(B,X)N1

F(B,X) which it is called the first cohomology group of B over F with coefficients in X.

26

Arens regularity of real Banach algebras and (−1)-weak amenability of second dual of real Banach algebras

A Banach algebra B over F is called amenable if H1F(B,X) = 0 for all Banach B-module X over F.

This concept was first introduced by Johnson in [9]. The notion of weak amenability was first introduced byBade, Curtis and Dales for commutative Banach algebras in [3] and later defined for Banach algebras, notnecessarily commutative, by Johnson in [10]. In fact, a Banach algebra B over F is called weakly amenableif H1

F(B,B∗) = 0.Let B be a Banach algebra over F. For each (λ,Λ) ∈ B∗ ×B∗∗ the F-valued functions λ ·Λ and Λ · λ on

B are defined by

(λ · Λ)(a) = Λ(a · λ) (a ∈ B),

(Λ · λ)(a) = Λ(λ · a) (a ∈ B).

Then λ · Λ ∈ B∗, ‖λ · Λ‖op ≤ ‖λ‖op‖Λ‖op, Λ · λ ∈ B∗ and ‖Λ · λ‖op ≤ ‖Λ‖op‖λ‖op. For each Λ,Γ ∈ B∗∗, theF-valued functions ΛΓ and Λ4Γ on A∗ are defined by

(ΛΓ)(λ) = Λ(Γ · λ) (λ ∈ A∗),(Λ4Γ)(λ) = Γ(λ · Λ) (λ ∈ A∗).

Then ΛΓ ∈ B∗∗, ‖ΛΓ‖op ≤ ‖Λ‖op‖Γ‖op, Λ4Γ ∈ B∗∗ and ‖ΛΓ‖op ≤ ‖Λ‖op‖Γ‖op. Moreover, B∗∗ is aBanach algebra over F with respect to either of the products and 4 and with the operator norm ‖ · ‖op.These products are called the first and second Arens products on B∗∗, respectively. The Banach algebraB over F is called Arens regular if two products and 4 coincide on B∗∗. For the general theory ofArens products, see [3, 5, 21], for example. For the product on B∗∗ one can show that B∗ is a BanachB∗∗-module over F if and only if the following statements hold:

(i) (Λ · λ) · Γ = Λ · (λ · Γ) for all (Λ, λ,Γ) ∈ B∗∗ ×B∗ ×B∗∗,

(ii) λ · (ΛΓ) = (λ · Λ) · Γ for all (λ,Λ,Γ) ∈ B∗ ×B∗∗ ×B∗∗,

(iii) (ΛΓ) · λ = Λ · (Γ · λ) for all (Λ,Γ, λ) ∈ B∗∗ ×B∗∗ ×B∗.

definition 1.1. Let (B, ‖ · ‖) be a Banach algebra over F and × be one of the Arens products and 4 onB∗∗. We say that B∗∗ (with the product ×) is (−1)-weakly amenable if B∗ is a Banach B∗∗-module over Fand H1

F(B∗∗, B∗) = 0.

Medghalchi and Yazdanpanah introduced the concept of (−1)-weak amenability for Banach algebras in[11] and obtained some results in this area. Eshaghi Gordji, Hosseinioun and Valadkhani in [6] gave someexamples of complex Banach algebras that their second duals which are and some others which are not(−1)-weakly amenable. Hosseinioun and Valadkhani obtained interesting results in (−1)-weak amenabilityof complex Banach algebras in [7, 8].

Let E be a real linear space (real algebra, respectively). A complex linear space (complex algebra,respectively) EC is called a complexification of E if there exists an injective real linear map (real algebrahomomorphism, respectively) J : E −→ EC such that EC = J(E)⊕ iJ(E).

If X is a real linear space, then X× X with the additive operation and scalar multiplication defined by

(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2) (x1, x2, y1, y2 ∈ X),

(α+ iβ)(x, y) = (αx− βy, αy + βx) (α, β ∈ R, x, y ∈ X),

is a complexification of X with respect to the injective linear map J : X −→ X × X defined by J(x) =(x, 0), x ∈ X.

If A is a real algebra, then A×A with the algebra operations

(a1, b1) + (a2, b2) = (a1 + a2, b1 + b2) (a1, a2, b1, b2 ∈ A),

(α+ iβ)(a, b) = (αa− βb, αb+ βa) (α, β ∈ R, a, b ∈ A),

(a1, b1)(a2, b2) = (a1a2 − b1b2, a1b2 + b1a2) (a1, b1, a2, b2 ∈ A),

27

D. Alimohammadi, H. Alihoseini

is a complexification of A with the injective real algebra homomorphism J : A −→ A × A defined byJ(a) = (a, 0), a ∈ A.

It is known [4, Proposition I.1.13] that if (E, ‖ · ‖) is a real normed algebra (real normed space, respec-tively), then there exists an algebra norm (a norm, respectively) ‖| · ‖| on E × E satisfying ‖|(a, 0)‖| = ‖a‖for all a ∈ E and

max‖a‖, ‖b‖ ≤ ‖|(a, b)‖| ≤ 2 max‖a‖, ‖b‖,for all a, b ∈ E.

definition 1.2. Let (E, ‖·‖) be a real normed linear space (real normed algebra, respectively), let a complexlinear space (algebra, respectively) EC be a complexification of E with respect to an injective real linearmap (real algebra homomorphism, respectively) J : E −→ EC and let ‖| · ‖| be a norm (an algebra norm,respectively) on EC. We say that ‖| · ‖| satisfies the (∗) condition if there exist positive constants k1 and k2

such that

max‖a‖, ‖b‖ ≤ k1‖|J(a) + iJ(b)‖| ≤ k2 max‖a‖, ‖b‖,for all a, b ∈ E.

Note that the (∗) condition implies that (E, ‖ ·‖) is a Banach space (Banach algebra, respectively) if andonly if (EC, ‖| · ‖|) is Banach space (Banach algebra, respectively). Moreover, the existence of a norm (analgebra norm, respectively) ‖ · ‖| on EC satisfying the (∗) condition guarantees by [4, Proposition I.1.13].

It is shown [1] that if (A, ‖ · ‖) is a real Banach algebra and if ‖| · ‖| is an algebra norm on complexalgebra A×A satisfying

max‖a‖, ‖b‖ ≤ k1‖|(a, b)‖| ≤ k2 max‖a‖, ‖b‖for some positive constants k1 and k2 and for all a, b ∈ A, then

(i) A is amenable if and only if A×A is amenable [1, Theorem 2.4].

(ii) A is weakly amenable if and only if A×A is weakly amenable [1, Theorem 2.5].

In Section 2 we assume that (A, ‖·‖) is a real Banach algebra, a complex algebra AC is the complexificationof A with respect to an injective real algebra homomorphism J : A −→ AC, ‖| · ‖| is an algebra norm on ACsatisfying the (∗) condition and (AC)∗ is the dual space of (AC, ‖| · ‖|). We first show that A is Arens regularif and only if AC is Arens regular. Next we prove that A∗ is a real Banach A∗∗-module if and only if (AC)∗

is a complex Banach (AC)∗∗-module. Moreover, we show that if A is a real Banach algebra such that A∗ is areal Banach A∗∗-module, then A∗∗ is (−1)-weakly amenable if and only if (AC)∗∗ is (−1)-weakly amenable.

2 Main results

We first give some lemmas which they will use in the sequel to prove of the main results.

lemma 2.1. Let (X, ‖ · ‖) be a real Banach space, let XC be a complexification of X with respect to aninjective real linear map J : X −→ XC, let ‖| · ‖| be a norm on XC satisfying the (∗) condition with respectto positive constants k1 and k2 and let (XC)∗ be the dual space of the complex Banach space (XC, ‖| · ‖|).

(i) Let ϕ ∈ X∗ and define the map ϕC : XC −→ C by

ϕC(J(x) + iJ(y)) = ϕ(x) + iϕ(y) (x, y ∈ X).

Then ϕC(J(x)) = ϕ(x) for all x ∈ X, ϕC ∈ (XC)∗, ‖ϕC‖op ≤ 2k1‖ϕ‖op and ‖ϕ‖op ≤ k2k1‖ϕC‖op.

(ii) Let λ ∈ (XC)∗ and define the map λR : X −→ R by

λR(x) = Reλ(J(x)) (x ∈ X).

Then λR ∈ X∗ and ‖λR‖op ≤ k2k1‖λ‖op.

28


(iii) Let λ ∈ (XC)∗ and define the map λI : X −→ R by

λI(x) = Imλ(J(x)) (x ∈ X).

Then λI ∈ X∗ and ‖λI‖op ≤ k2k1‖λ‖op.

lemma 2.2. Let (X, ‖ · ‖) be a real Banach space, let XC be a complexification of X with respect to aninjective real linear map J : X −→ XC, let ‖| · ‖| be a norm on XC satisfying (∗) condition with respectto positive constants k1 and k2 and let (XC)∗ be the dual space of the complex Banach space (XC, ‖| · ‖|).Define the map J1 : X∗ −→ (XC)∗ by

J1(ϕ) = ϕC , (ϕ ∈ X∗). (2)

Then:

(i) J1(ϕ)(J(x) + iJ(y)) = ϕ(x) + iϕ(y) for all ϕ ∈ X∗ and x, y ∈ X.

(ii) J1 is a real linear map from X∗ into (XC)∗.

(iii) If λ ∈ (XC)∗, then λ = J1(λR) + iJ1(λI).

(iv) J1 is injective and (XC)∗ = J1(X∗)⊕ iJ1(X∗).

(vii) (XC)∗ is a complexification of X∗ with respect to the map J1 : X∗ −→ (XC)∗ defined by (2) and

max‖ϕ‖op, ‖ψ‖op ≤k2

k1‖J1(ϕ) + iJ1(ψ)‖op

≤ 4k2 max‖ϕ‖op, ‖ψ‖op,

for all ϕ,ψ ∈ X∗.

lemma 2.3. Let (X, ‖ · ‖) be a real Banach space, let XC be a complexification of X with respect to aninjective real linear map J : X −→ XC, let ‖| · ‖| be a norm on XC satisfying (∗) condition with positiveconstants k1 and k2 and let (XC)∗ be the dual space of (XC, ‖| · ‖|). Define the map J2 : X∗∗ −→ (XC)∗∗ by

J2(Φ) = ΦC (Φ ∈ X∗∗). (3)

Then

(i) J2(Φ)(J1(ϕ) + iJ1(ψ)) = Φ(ϕ) + iΦ(ψ) for all Φ ∈ X∗∗ and ϕ,ψ ∈ X∗.

(ii) J2 is a real linear map from X∗∗ into (XC)∗∗ .

(iii) If Λ ∈ (XC)∗∗, then Λ = J2(ΛR) + iJ2(ΛI).

(iv) J2 is injective and (XC)∗∗ = J2(X∗∗)⊕ iJ2(X∗∗).

(v) (XC)∗∗ is a complexification of X∗∗ with respect to the map J2 : X∗∗ −→ (XC)∗∗ defined by (3) and

max‖Φ‖op, ‖Ψ‖op ≤ 4k1‖J2(Φ) + iJ2(Ψ)‖op≤ 16k2 max‖Φ‖op, ‖Ψ‖op,

for all Φ,Ψ ∈ X∗∗.

(vi) J2 πX = πXC J , whenever πY : Y −→ Y ∗∗ is the natural embedding Y in Y ∗∗ defined by

πY (y)(λ) = λ(y) (y ∈ Y, λ ∈ Y ∗).

29

D. Alimohammadi, H. Alihoseini

lemma 2.4. Let (A, ‖ · ‖) be a real Banach algebra, let AC be a complexification of A with respect to aninjective real algebra homomorphism J : A −→ AC, let ‖| · ‖| be an algebra norm on AC satisfying the (∗)condition and let (AC)∗ be the dual space of (AC, ‖| · ‖|).

(i) If a ∈ A and ϕ ∈ A∗, then

J1(a · ϕ) = J(a) · J1(ϕ), J1(ϕ · a) = J1(ϕ) · J(a).

(ii) If ϕ ∈ A∗ and Φ ∈ A∗∗, then

J1(ϕ · Φ) = J1(ϕ) · J2(Φ), J1(Φ · ϕ) = J2(Φ) · J1(ϕ).

(iii) If Φ,Ψ ∈ A∗∗, then

J2(ΦΨ) = J2(Φ)J2(Ψ), J2(Φ4Ψ) = J2(Φ)4J2(Ψ).

(iv) If Λ ∈ (AC)∗∗ and λ ∈ (AC)∗, then

Λ · λ = J1(ΛR · λR − ΛI · λI) + iJ1(ΛR · λI + ΛI · λR),

λ · Λ = J1(λR · ΛR − λI · ΛI) + iJ1(λR · ΛI + λI · ΛR)

theorem 2.5. Let (A, ‖ · ‖) be a real Banach algebra, let AC be a complexification of A with respect to aninjective real algebra homomorphism J : A −→ AC, let ‖| · ‖| be an algebra norm on AC satisfying the (∗)condition and let (AC)∗ be the dual space of (AC, ‖| · ‖|). Then:

(i) A is reflexive if and only if AC is reflexive.

(ii) A is Arens regular if and only if AC is Arens regular.

theorem 2.6. Let (A, ‖ · ‖) be a real Banach algebra, let AC be a complexification of A with respect to aninjective real algebra homomorphism J : A −→ AC, let ‖| · ‖| be an algebra norm on AC satisfying the (∗)condition and let (AC)∗ be the dual space of (AC, ‖| · ‖|). Then A∗ is a real Banach A∗∗-module if and onlyif (AC)∗ is a complex Banach (AC)∗∗-module.

We now discuss the relationship between the (−1)-weak amenability of A∗∗ and (−1)-weak amenabilityof (AC)∗∗. For this purpose we need the following lemma.

lemma 2.7. Let (A, ‖ · ‖) be a real Banach algebra, let AC be a complexification of A with respect to aninjective real algebra homomorphism J : A −→ AC, let ‖| · ‖| be an algebra norm on AC satisfying (∗)condition and let (AC)∗∗ be the second dual of (AC, ‖| · ‖|). Suppose that A∗ is a real Banach A∗∗-module.Then:

(i) If d ∈ Z1R(A∗∗, A∗) and Φ ∈ A∗∗, then J1(d(Φ)) ∈ (AC)∗.

(ii) If d ∈ Z1R(A∗∗, A∗) then ∆d ∈ Z1

C((AC)∗∗, (AC)∗), where the map ∆d : (AC)∗∗ −→ (AC)∗ is defined by

∆d(J2(Φ) + iJ2(Ψ)) = J1(d(Φ)) + iJ1(d(Ψ)), (Φ,Ψ ∈ A∗∗). (4)

(iii) The map JZ : Z1R(A∗∗, A∗) −→ Z1

C((AC)∗∗, (AC)∗) defined by

JZ(d) = ∆d (d ∈ Z1R(A∗∗, A∗) (5)

is an injective real linear map.

(iv) The complex linear space Z1C((AC)∗∗, (AC)∗) is a complexification of the real linear space Z1

R(A∗∗, A∗)with respect to the injective linear map JZ .

30


(v) If ϕ ∈ A∗, then JZ(δϕ) = δJ1(ϕ).

(vi) If λ ∈ (AC)∗, then δλ = JZ(δλR) + iJZ(δλI ).

(vii) H1R(A∗∗, A∗) = 0 if and only if H1

C((AC)∗∗, (AC)∗) = 0.

theorem 2.8. Let (A, ‖ · ‖) be a real Banach algebra, let AC be a complexifiction of A with respect to aninjective real algebra homomorphism J : A −→ AC, let |‖ · ‖| be an algebra norm on AC satisfying the (∗)condition, and let (AC)∗ be the dual space of (AC, ‖| · ‖|). Then A∗∗ is (−1)-weakly amenable if and only if(AC)∗∗ is (−1)-weakly amenable.

References

[1] D. Alimohammadi and T. G. Honary, Contractibility, amenability and weak amenability of real Banachalgebras, J. Aanalysis, (9)(2001), pp. 69-88.

[2] R. Arens, The adjoint of a bilinear operation, Proc. Math. Amer. Soc. 2(1951), pp. 839-848.

[3] W. G. Bade, P. C. Curtis, Jr and H. G. Dales, Amenability and weak amenability for Beurling andLipschitz algebras, Proc. London Math. Soc. 55(1987), pp. 359-377.

[4] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer Verlag, New York, 1973.

[5] J. Duncan and S. A. R. Hosseinioun, The second dual of a Banach algebra, Proc. Roy. Soc. EdinburgSect. A. 84 (1979), pp. 309-325.

[6] M. Eshaghi Gordji, S. A. R. Hosseinioun, and A. Valadkhani, On (-1)-weak amenability of Banachalgebras, Math. Reports, 15(65), (2013), pp. 271-279.

[7] S. A. R. Hosseinioun and A. Valadkhani, (-1)-Weak amenability of unitized Banach algebras, Europ.J. Pure Appl. Math. 9(2016), pp. 231-239.

[8] S. A. R. Hosseinioun and A. Valadkhani, Weak and (-1)-weak amenability of second dual of Banachalgebras, Int. J. Nonlinear Anal. Appl. 7(2)(2016), pp. 39-48.

[9] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972).

[10] B. E. Johnson, Derivations from L1(G) into L1(G) and L∞(G), Proc. International conference onHarmonic Analysis, Luxembourg, (Lecture note in Math. Springer-Verlag), 1359 (1987), pp. 191-198.

[11] A. Medghalchi and T. Yazdanpanah, Problems concerning n-weak amenability of a Banach algebra,Czechoslovak Math. J. 55(4)(2005), pp. 863-876.

[12] T. W. Palmer, Banach Algebras, the General Theory of *-Algebras, Vol. 1: Algebras and BanachAlgebras, Cambridge University Press, Cambridge, 1994.

31

Generalization of the Hausdorff-Young inequality for Pseudo- integral

A. Alishahi*

(Payame Noor University, Tehran, Iran)

Abstract: In this paper,we continue the study of pseudo-integral for Lp-spaces that was investigatedby Endre Pap, Mirjana Strboja and Imer Rudas in 2014. In the sequel, by using monotone and continuousfunction g we generalizations of the Hausdorff-Young inequality for the pseudo-integral. There are con-sidered two cases of the real semiring with pseudo-operation and proven generalizations of the Holder andMinkowski’s inequalities for pseudo-operation.

keywords. pseudo-integral, Hausdorff-Young inequality, pseudo-operation

subject. 13D45, 39B42

1 Introduction

Pseudo-analysis is ageneralization of the classical analysis, where instead of the field of real numbers a semir-ing is taken on a real interval [a, b] ⊂ [0,∞] endowed with psedo-addition ⊕ and with pseudo-multiplication, see [1], [2].The full order on [a, b] will be denoted by . A binary operation ⊕ on [a, b] is pseudo-additionif it is commutative, non-decreasing (with respect to ), associative and with a zero (neutral) element denoteby 0. Let [a, b]+ = x;x ∈ [a, b], 0 x. A binary operation on [a, b] is pseudo-multiplication if it iscommutative, positively non-decreasing, i.e., x y implies x z y z for all z ∈ [a, b]+, associative andwith a unite element 1 ∈ [a, b], i.e., for each x ∈ [a, b], 1 x = x. We assume also 0 x = 0 and that isdistributive over ⊕, i.e.,

x (y ⊕ z) = (x y)⊕ (x z)The structure ([a, b],⊕,) is a semiring. In this paper we will consider only semirings with with the followingcontinuous operationand where the boundary elements of the interval [a, b] are the neutral elements of thepseudo-operation:

Case I: The pseudo-addition is idempotent operation and the pseudo-multiplication is not.

(a) x ⊕ y = sup(x, y), is orbitrary not idempotent pseudo-multiplication on the interval [a, b]concelative on ]a, b[2. We have 0 = a and the idempotent operation supinduces a full order inthe following way: x y if and only if sup(x, y) = y. Moreover, the pseudo multiplication isgenerated by an increasing bijection g : [a, b]→ [0,∞], x y = g−1(g(x).g(y)) (this result followsfrom [4], see also [3]).

(b) x ⊕ y = inf(x, y), is orbitrary not idempotent pseudo-multiplication on the interval [a, b]concelative on ]a, b[2. We have 0 = b and the idempotent operation supinduces a full order inthe following way: x y if and only if inf(x, y) = y. Moreover, the pseudo multiplication isgenerated by an decreasing bijection g : [a, b] → [0,∞], x y = g−1(g(x).g(y)). Observe that1 = g−1(1).

32


Case II: The pseudo-operations are defined by monotone and continuous function g : [a, b] → [0,∞], i.e.,pseudo-operations are given with x⊕ y = g−1(g(x) + g(y)) and x y = g−1(g(x) g(y)).

If x y = g−1(g(x) g(y)), then x(p) = x · · · x︸︷︷︸

p

= g−1(gp(x)). The pseudo-integral of the function

f : X → [a, b] is defined by∫ ⊕X f dm = g−1

(∫X(g f)d(g m)

).

2 Some properties of the space Lp⊕

For basic definitions of pseudo-Lp and Lp⊕ see [5].

definition 2.1. Let (fn)n∈N be a sequence in Lp⊕.⊕∞

n=1 fn converges to s in the mean of order p(p > 0) ifwe have limn→∞Dp⊕(sn, s) = 0 where sn =

⊕nk=1 fn.

definition 2.2. Let (fn)n∈N be a sequence in Lp⊕.⊕∞

n=1 fn fundamentally converges in the mean of order

p(p > 0) if we have limn→∞(∫ ⊕

X

(⊕nk=m+1 fk

)(p) dm

)( 1p

)

= 0.

theorem 2.3. Let 1 ≤ p <∞. A pseudo- series⊕∞

n=1 fn converges to s in the mean of order p in Lp⊕ iff itfundamentally converges in the mean of order p in Lp⊕.

Proof. Assume that 1 ≤ p < ∞ and a pseudo- series⊕∞

n=1 fn converges to s in Lp⊕. Since the sequence(sn)n∈N converges to s in the mean of order p then by (theorem 22, [5]) the sequence (sn)n∈N fundamentallyconverges in the mean of order p in Lp⊕ and we have limn→∞Dp⊕(sn, sm) = 0. So

0 = limn→∞

Dp⊕(sn, sm) = limm,n→∞

(∫ ⊕

Xd⊕(sn, sm))

(p) dm

)( 1p

)

= limm,n→∞

(g−1

∫

X|g(sn)− g(sm|(p)d(g m)

)( 1p

)

= limm,n→∞

(g−1

∫

X(g(

n⊕

k=m+1

fk))pd(g m)

)( 1p

)

= limm,n→∞

(∫ ⊕

X(

n⊕

k=m+1

fk)(p) dm

)( 1p

)

The following theorem holds for case (II).

theorem 2.4. If f ∈ L1⊕ ∩ L∞⊕ , 1 < p < ∞ and generator g : [a, b] → [0,∞] of the pseudo-addition ⊕ and

pseudo-multiplication is an increasing and continuous, then f ∈ Lp⊕.

Proof. Let p ∈ (1,∞). Since f ∈ L1⊕ we have

∫ ⊕X f dm < b. Assume that f ∈ L∞⊕ , then f < b m-a.e.Using

33

A. Alishahi

the properties of the pseudo-integral and pseudo-power we obtain

(∫ ⊕

Xf

(p) dm

)( 1p

)

=

∫ ⊕

X(f · · · f)︸︷︷︸

p−times

dm

( 1p

)

=

∫ ⊕

X(f · · · f)︸︷︷︸

(p−1)−times

f dm

( 1p

)

<

∫ ⊕

X(b · · · b)︸︷︷︸(p−1)−times

f dm

( 1p

)

m− a.e

<

(b · · · b)︸︷︷︸

(p−1)−times

∫ ⊕

Xf dm

( 1p

)

m− a.e

<

(b · · · b b)︸︷︷︸

p−times

( 1p

)

=(b(p))( 1

p)

= b

So f ∈ Lp⊕.

proposition 2.5. Let ([a, b],⊕,) be a semiring either from case (I) or from (II). If m(X) < ∞ and1 ≤ p < q <∞ then

(∫ ⊕X f

(p) dm

)( 1p

)

≤ m(X)

1p− 1q

(∫ ⊕X f

(q) dm

)( 1q

)

Proof. Let m(X) <∞ and 1 ≤ p < q <∞. Taking f(p) instead of f and 1 instead of h in Holder inequality

[5] depending on the type of the semiring, we obtain

(∫ ⊕

X(f

(p) 1) dm

)( 1p

)

≤

(∫ ⊕

X(f

(p) )

( qp

)

dm)( 1

q)

(∫ ⊕

X(1)

( qq−p )

dm)( q−p

pq)

= m(X)( q−ppq

)

(∫ ⊕

Xf

(q) dm

)( 1q

)

3 Main results

In this section we prove the Hausdorff-Young inequality. Let f(n) =∫X fϕndµ, (n = 1, 2, · · · ), ϕn be an

bounded sequence in L∞(µ) and f ∈ Lp(µ), 1 ≤ p ≤ 2.

theorem 3.1. For a given measurable space (X,A), let f, ϕn : X → [a, b] be measurable functions andlet a generator g : [a, b] → [0,∞] of the pseudo-addition ⊕ and pseudo-multiplication be an increasingfunction. Then for σ −⊕- measure m it holds:

34


(∫ ⊕X f

(q) dm

)( 1q

)

≤M ( 2−p

p)

(∫ ⊕

X f(p) dm

)( 1p

)

Proof. We apply the classical Hausdorff-Young inequality on compositions g f and g fand then we obtain

(∫X(g f)qdg m

) 1q ≤ g(M)

2−pp(∫X(g f)pdg m

) 1p .

Since the function g is increasing function, then g−1 is also increasing function and we have

g−1

(∫

X(g f)qdg m

) 1q

≤ g−1

(g(M)

2−pp

(∫

X(g f)pdg m

) 1p

)

i. e.,

g−1

(∫

X(g f)qdg m

) 1q

= g−1

(∫

Xgg−1(g f)qdg m

) 1q

= g−1

(∫

Xg(f

(q) )dg m

) 1q

= g−1

(gg−1

∫

Xg(f

(q) )dg m

) 1q

= g−1

(g

∫

Xf

(q) dm

) 1q

=

(∫ ⊕

Xf

(q) dm

)( 1q

)

.

For the right side of the inequality:

g−1

(g(M)

2−pp

(∫

X(g f)pdg m

) 1p

)= g−1

(g(g−1(g(M)

2−pp ))g

(g−1

∫

X(g f)pdg m

) 1p

)

=(g−1(g(M)

2−pp ))(g−1

(∫

X(g f)pdg m)

) 1p

)

= M( 2−pp

)

(g−1

(∫

Xg(g−1(g f)p)dg m)

) 1p

)

= M( 2−pp

)

(g−1

(∫

Xg(fp)dg m)

) 1p

)

= M( 2−pp

)

(g−1

(g

(g−1

∫

Xg(fp)dg m)

)) 1p

)

= M( 2−pp

)

(g−1

(g

(∫ ⊕

Xfp dm

)) 1p

)

= M( 2−pp

)

(∫ ⊕

Xfp dm

)( 1p

)

Now we shall give a generalization of the classical Hoder and Minkowski’s inquality.

35

A. Alishahi

theorem 3.2. Let p and q be conjugate exponents, 1 < p <∞, xi, yi ≥ 0, i = 1, 2, · · · , n and let a generatorg : [a, b]→ [0,∞] of the pseudo-addition and the pseudo-multiplication be an increasing functions. Then

(i)⊕n

i=1 ≤(⊕n

i=1 x(p)i)( 1

p)

(⊕n

i=1 y(q)i)( 1

q)

(ii)(⊕n

i=1(xi ⊕ yi)(p))( 1

p)

≤(⊕n

i=1 x(p)i)( 1

p)

⊕(⊕n

i=1 y(p)i)( 1

p)

Proof. (i) We apply the classical Hoder inquality and then we obtian

∑ni=1 g(xi)g(yi) ≤ (

∑ni=1 g(xi)

p)1p (∑n

i=1 g(yi)q)

1q .


g−1

(n∑

i=1

g(xi)g(yi)

)≤ g−1

(

n∑

i=1

g(xi)p

) 1p(

n∑

i=1

g(yi)q

) 1q

.

Then

g−1

(n∑

i=1

g(xi)g(yi)

)= g−1

(n∑

i=1

gg−1(g(xi)g(yi))

)

= g−1

(n∑

i=1

g(xi yi))

= g−1

(g

(g−1

(n∑

i=1

g(xi yi))))

= g−1

(g

(n⊕

i=1

(xi yi)))

=n⊕

i=1

(xi yi)

For the right side of the inequality holds:

g−1

( n∑i=1

g(xi)p

) 1p(

n∑i=1

g(yi)q

) 1q

= g−1

g

g−1

(n∑i=1

g(xi)p

) 1p

g

g−1

(n∑i=1

g(yi)q

) 1q

= g−1

(n∑i=1

g(xi)p

) 1p

g−1

(n∑i=1

g(yi)q

) 1q

= g−1

(n∑i=1

g(g−1(g(xi)p))

) 1p

g−1

(n∑i=1

g(g−1(g(yi)q))

) 1q

= g−1

(n∑i=1

g(x(p)i )

) 1p

g−1

(n∑i=1

g(y(q)i )

) 1q

= g−1

(g

(g−1

(n∑i=1

g(x(p)i )

))) 1p

g−1

(g

(g−1

(n∑i=1

g(y(q)i )

))) 1q

=

(n⊕i=1

x(p)i

)( 1p

)

(n⊕i=1

y(q)i

)( 1q

)

36


(ii) We apply the classical Minkowski’s inquality for g(xi), g(yi)and then we obtian

(∑n

i=1(g(xi) + g(yi))p)

1p ≤ (

∑ni=1 g(xi)

p)1p + (

∑ni=1 g(yi)

p)1p .


g−1

(n∑

i=1

(g(xi) + g(yi))p

) 1p

≤ g−1

(

n∑

i=1

g(xi)p

) 1p

+

(n∑

i=1

g(yi)p

) 1p

.

Then simillary part (i) we have

(⊕ni=1(xi ⊕ yi)(p)

)( 1

p)

≤(⊕n

i=1 x(p)i)( 1

p)

⊕(⊕n

i=1 y(p)i)( 1

p)

.

So completes the proof.

References

[1] E. Pap, g-calculus, Univ. u Novon Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat.23(1)(1993) 145-156.

[2] E. Pap, Null-Additive Set Functions, Kluwer Academic Publishers, Dordrecht, Boston, London, 1995.

[3] J. C. Fodor, R. R. Yager, A. Ryballov, Structure Of Uninorm, Int. J. Uncertain. Fuzziness Knowledge-Based Syst. 5(1997)411-427.

[4] E.P Klement, R. Mesiar, E. Pap, On The Relationship Of Associative Compensatory Operators ToTriangular norms and Conorms, Int. J. Uncertain. Fuzziness Knowledge-Based Syst. 4(2) (1996) 129-144.

[5] E. Pap, M. Strboja, I. Rudas, Pseudo-Lp Space and Convergence, Fuzzy Sets and Systems 238 (2014)113-128.

37

Optimal dual frames in Hilbert spaces

F. Arabyani Neyshaburi*, A. A. Arefijamaal and Gh. Sadeghi

(Hakim Sabzevari University, Sabzevar, Iran)

Abstract: One of the most precious problems in frame theory, which is firstly introduced by Han etal., is the optimal dual problem, finding the best dual frame that minimizes the reconstruction errors whenerasures occur.We consider this problem using a new point of view. To this end, we characterize extreme points in theset of all optimal duals for one erasure. Moreover, we introduce the notion of numerically optimal dualsand give several equivalent conditions for which the canonical dual and alternate duals are spectrally andnumerically optimal duals.

keywords. Frames, optimal duals, extreme points, spectrally optimal duals, numerically optimal duals.

subject. 42C15, 46C05, 42C40.

1 Introduction

In signal transmission, coding theory and many applications a signal f encode by the analysis operator θF fand then receiver decode f by the synthesis operator of a dual frame G of F such as θ∗GθF f . In many ofthese transmissions usually a part of the data vectors are erased, and we need to perform the reconstructionby using the partial information at hand. Optimal dual problem deal with minimizing of the maximal errorsunder erasures. This concept was considered applying two approaches, first by J. Lopez and D. Han in [6] andthen by S. Pehlivan, D. Han and R. Mohapatra in [7]. In this paper, we first present a new characterizationof optimal duals in terms of the notion introduced in [6]. Then we introduce a new measurement for erroroperator and compare it with approaches considered in [5, 6, 7].

Let H be a separable Hilbert space and I a countable index set. A sequence F := fii∈I ⊆ H is calleda frame for H if there exist the positive constants 0 < A ≤ B <∞ such that

A‖f‖2 ≤∑

i∈I|〈f, fi〉|2 ≤ B‖f‖2, (f ∈ H). (6)

The constants A and B are called the frame bounds. If A and B can be chosen such that A = B, the frameF is called a tight frame, and in the case of A = B = 1 it is a Parseval frame. For a frame fii∈I , thesynthesis operator TF : l2 → H is defined by TF ci =

∑i∈I cifi. If fii∈I is a frame, then SF = TFT

∗F

where T ∗F : H → l2 the adjoint of T , given by T ∗F f = 〈f, fi〉i∈I , is called the analysis operator . A frameG := gii∈I ⊆ H is called a dual for fii∈I if TGT

∗F = IH. A special dual frame is S−1

F fii∈I , whichis called the canonical dual of F [2]. It is well known that gii∈I is a dual of frame fii∈I if and only ifgi = S−1

F fi + ui, for all i ∈ I where uii∈I satisfies

∑

i∈I〈f, ui〉fi = 0, (f ∈ H). (7)

38


In this paper we only consider finite frames in Hilbert spaces. Consider Ik = 1, ..., k and let F =fii∈Ik be a frame for n-dimensional Hilbert space Hn, in this case we call F a (k, n)-frame. If G = gii∈Ikis a dual of F and J ⊂ Ik, then the error operator EJ is defined by

EJf =∑

i∈J〈f, fi〉gi = (TGDT

∗F )f, (f ∈ H)

where D is k × k diagonal matrix with dii = 1 for i ∈ J and 0 otherwise. Let

dr(F,G) = max‖TGDT ∗F ‖ : D ∈ Dr = max‖EJ‖ : card(EJ) = r, (8)

in which the norm used in (8) is the operator norm, and Dr is the set of all k × k diagonal matrices with r1′s and n− r 0′s. Then dr(F,G) is the largest possible error when r-erasures occur. Indeed, G is called anoptimal dual frame of F for 1-erasure if

d1(F,G) = min d1(F, Y ) : Y is a dual of F . (9)

Inductively, a dual frame G is called an optimal dual of F for r-erasures if it is optimal for (r − 1)-erasuresand

dr(F,G) = min dr(F, Y ) : Y is a dual of F .

In [6], it was shown that, if F = fii∈Ik is a frame for Hn with fi 6= 0 for all i, then the set of optimalduals of F for any r-erasures is a non-empty, compact and convex subset of Hn. In this paper, we alsoconsider frames with non-zero elements. Hence, if a dual is the unique optimal dual of F for 1-erasurethen it is the unique optimal dual for any r-erasures. Now, let F = fii∈Ik be a frame of Hn, putc = max‖S−1

F fi‖‖fi‖ : 1 ≤ i ≤ k , Λ1 = 1 ≤ i ≤ k : ‖S−1F fi‖‖fi‖ = c and Λ2 = 1, ..., k \ Λ1. Also

let Hi = spanfj : j ∈ Λi, for i = 1, 2. Using these notations, the authors in [5] obtained some conditionsunder which the canonical dual is either the unique optimal dual, a non-unique optimal dual, or a nonoptimal dual for 1-erasure.

theorem 1.1. Let F = fii∈Ik be a frame of Hn. The following are equivalent;

(i) The canonical dual S−1F fii∈Ik is the unique optimal dual for 1-erasure.

(ii) H1 ∩H2 = 0 and fii∈Λ2 is linearly independent.

proposition 1.2. Let F = fii∈Ik be a frame of Hn. Also, let fii∈Λ1 be linearly independent. Then

(i) If H1∩H2 = 0, the canonical dual is an optimal dual frame but not the only optimal dual for 1-erasure.

(ii) If there exists a sequence of scalars cii∈Ik such that∑

i∈Ik cifi = 0 and ci 6= 0, for all i ∈ Λ1. Then thecanonical dual is not optimal dual frame of F for 1-erasure.

2 Main results

In this section, we survey some conditions under which an alternate dual frame is an optimal dual for 1-erasure. Moreover, we characterize all extreme points in the set of optimal dual frames of F . The importanceof this work comes from the fact that the set of extreme points in a compact convex set is the smallest subsetfor identifying and constructing of the set. First, we need to some notations.

Let gii∈Ik be a dual frame of fii∈Ik , then we consider cg = max‖gi‖‖fi‖ : i ∈ Ik, Λg1 = i ∈ Ik :‖gi‖‖fi‖ = cg and Λg2 = Ik \ Λg1. Also let Hgi = spanfj : j ∈ Λgi , for i = 1, 2. Also, we denote the set ofall duals by DF , and optimal duals by ODF .

proposition 2.1. [1] Let F = fii∈Ik be a frame of Hn and G = gii∈Ik be an its dual

(i) If fii∈Λg1is linearly independent and Hg1 ∩ Hg2 = 0. Then gii∈Ik is an optimal dual frame of F for

1-erasure, but not the only optimal one.

39

F. Arabyani Neyshaburi, A. A. Arefijamaal and Gh. Sadeghi

(ii) If gii∈Λg1is linearly independent and there exists a sequence of scalars cii∈Ik such that

∑i∈Ik cifi = 0

and ci 6= 0, for all i ∈ Λg1. Then G is not an optimal dual frame of F for 1-erasure (and so for any r-erasures).

It is known that ODF is a non-empty convex compact space [6]. Hence, by Krein-Milman theoremextODF , the set of all extreme points of ODF , is non-empty. Moreover, ODF is the closed convex hull ofits extreme points, i.e., ODF = co(extODF ), see [4]. Recall that, for a convex subset Y of a vector spaceX, a point y ∈ Y is an extreme point of Y if there is no proper open line segment which contain y and liesentirely in Y [4]. In the next theorem we characterize all extreme points of ODF .

theorem 2.2. Let F = fii∈Ik be a frame for Hn and for 1-erasure G = gii∈Ik ∈ ODF . Then G is anextreme point of ODF if and only if fii∈Λg2

is linearly independent.

corollary 2.3. Let F = fii∈Ik be a frame for Hn and for 1-erasure G = gii∈Ik ∈ ODF . Then G is anextreme point of ODF if and only if there is no optimal dual frame hii∈Ik of F such that gi = hi, for alli ∈ Λg1.

Theorem 2.2 also shows that the canonical dual, which has minimal l2-norm between all duals of F , isnot necessarily an extreme point if it is an optimal dual. On the other hand, ODF is a non-empty andcompact set so it has some elements with maximal l2-norm. In the next proposition, which is obtained bythe elementary maximum principle [4], we observe that such elements are the extreme points of ODF .

proposition 2.4. Let F = fii∈Ik be a frame for Hn and G = gii∈Ik ∈ ODF has the maximal l2-normbetween all optimal duals of F . Then G is an extreme point of ODF .

In the sequel, we are going to survey the relation between optimal duals of equivalent frames.

theorem 2.5. Suppose that F = fii∈Ik is a frame for Hn and U ∈ B(Hn) is an invertible operator suchthat ‖U‖‖U−1‖ ≤ 1. Then, under 1-erasure gii∈Ik is an optimal dual of F if and only if Ugii∈Ik is anoptimal dual of (U∗)−1F := (U∗)−1fii∈Ik . Moreover, there is a one to one corresponding between optimalduals of F and UF .

3 Numerically and spectrally optimal duals

In [7] the spectral radius of the error operator as a new measure of optimality was considered to characterizespectrally 1-erasure optimal duals. More precisely, let F = fii∈Ik be a frame of Hn with a dual frameG = gii∈Ik . For every r, let

ρrF,G = maxρ(EΛ) : |Λ| = r.

and

ρrF = minρrF,G : G is a dual of F,

where ρ(EΛ) is the spectral radius of the operator EΛ. Then a dual frame G of F is called 1-erasure spectrallyoptimal if ρ1

F = ρ1F,G. Moreover G is called r-erasures spectrally optimal if it is (r − 1)-erasures spectrally

optimal and ρrF = ρrF,G. Moreover, in 1-erasure case we obtain ρ1F,G = max|〈gi, fi〉| : i ∈ Ik, see [7].

In this section, we introduce the notion of numerically optimal duals and present some sufficient condi-tions, and equivalent conditions where determine (unique) numerically optimal duals. Moreover, we studyon relations between optimal duals with different measurements.

definition 3.1. Let F = fii∈Ik be a frame of Hn. For every r, let

ωrF,G = maxω(EΛ) : |Λ| = r.

where ω(EΛ) is the numerical radius of the error operator, i.e.,

ω(EΛ) = sup|〈EΛf, f〉| : ‖f‖ = 1,

40


Moreover, put

ωrF = minωrF,G : G is a dual of F.Then a dual frame G = gii∈Ik of F is called 1-erasure numerically optimal if ω1

F = ω1F,G.

Also, for r > 1, the dual frame G is said to be r-erasures numerically optimal if it is (r − 1)-erasuresnumerically optimal and ωrF = ωrF,G. In this section, we denote the set of all spectrally optimal dualsof a frame F by SODF , and numerically optimal duals by NODF . Applying the fact that the mappingEΛf = 〈f, fi〉gi is rank one in 1-erasure case we obtain the next result.

lemma 3.2. Let F = fii∈Ik be a (k, n)-frame and G = gii∈Ik be a 1-erasure numerically optimal F .Then

ω1F,G = max

i∈Ik

|〈gi, fi〉|+ ‖fi‖‖gi‖2

. (10)

theorem 3.3. Let F = fii∈Ik be a Parseval frame of Hn. Then the following assertions are equivalent;

(i) The canonical dual is an optimal dual of F for 1-erasure.

(ii) The canonical dual is a numerically optimal dual of F for 1-erasure.

(iii) The canonical dual is a spectrally optimal dual of F for 1-erasure.

The above theorem for general frames necessarily does not hold.

example 3.4. Suppose

F =

1

1

,

1

−1

,

0

2

.

Then F is a frame for R2 and the set of all dual frames of F is

DF =

1

2+ α

1

6+ β

,

1

2− α

−1

6− β

,

−α

1

3− β

,

for all α, β ∈ R. Putting α = 0 and β = −0.01 we obtain a dual frame as follows

H =

1

2

1

6− 0.01

,

1

2

−1

6+ 0.01

,

0

1

3+ 0.01

,

consequently max3i=1 ‖hi‖‖fi‖ < max3

i=1 ‖S−1F fi‖‖fi‖ =

√5

3and

3maxi=1

|〈hi, fi〉|+ ‖fi‖‖hi‖2

<3

maxi=1

|〈S−1F fi, fi〉|+ ‖fi‖‖S−1

F fi‖2

=

√5 + 2

6.

So the canonical dual is neither optimal dual nor numerically optimal dual. However, we show that SODF =S−1

F F. To the contrary, let there exists an alternate dual frame G = gi3i=1 so that

3maxi=1|〈gi, fi〉| = max

|23

+ α+ β|, |23− α+ β|, |2

3− 2β|

≤ 3

maxi=1|〈S−1

F fi, fi〉| =2

3,

without loss of generality let β 6= 0. Summing up the phrases

|23

+ α+ β| ≤ 2

3, |2

3− α+ β| ≤ 2

3,

implies that−4

3≤ β < 0 and simultaneous |2

3− 2β| ≤ 2

3implies that 0 < β ≤ 2

3that is a contradiction so

the desired result follows.

41

F. Arabyani Neyshaburi, A. A. Arefijamaal and Gh. Sadeghi

corollary 3.5. Suppose F = fii∈Ik is a uniform tight frame of Hn. Then ODF = NODF ⊆ SODF .

In the following theorem we extend some part of Theorem 1.3 for any r-erasures.

theorem 3.6. Let F = fii∈Ik be a Parseval frame of Hn. Then the following assertions hold;

(i) For any r-erasures, if the canonical dual is spectrally optimal dual of F then it is numerically optimaldual of F .

(ii) For any r-erasures, if the canonical dual is numerically optimal dual of F then it is optimal dual of F .

Finally, we obtain the spectrally and numerically optimal duals of group representation frames for 1-erasure. Recall that a unitary representation π on a group G is a group homomorphism from G into thegroup of unitary operators on some Hilbert space H. Moreover, a unitary representation π for a group G iscalled a frame representation whenever there exists ϕ ∈ H so that π(g)ϕ : g ∈ G is a frame for H, [3]. Inthis case π(g)ϕ : g ∈ G is said to be a group representation frame. It is shown that the canonical dual isthe unique optimal dual for group representation frames, [6]. To show our result, we consider GDΓ as theset of all dual frames of Γ with the same group structure.

theorem 3.7. Let Γ = π(g)ϕ : g ∈ G be a group representation frame. Then for 1-erasure the followingshold;

(i) GDΓ ⊆ SODΓ.

(ii) The unique numerically optimal dual with the same group structure is the canonical dual.

References

[1] F. Arabyani Neyshaburi, A. Arefijamaal and Gh. Sadeghi, Extreme points and identification of optimalalternate dual frames, Linear Algebra Appl. 549 (2018), 123-135.

[2] O. Christensen, Frames and Bases: An Introductory Course, Birkhauser, Boston, 2008.

[3] D. Han, Classification of finite group-frames and super-frames, Canad. Math. Bull. 50 (2007), 85-96.

[4] R. Lay. Steven, Convex sets and their applications, John Wiley and Sons, New York, 1982.

[5] J. Leng, D. Han, Optimal dual frames for erasures II, Linear Algebra Appl. 435 (6) (2011), 1464-1472.

[6] J. Lopez, D. Han, Optimal dual frames for erasures, Linear Algebra Appl. 432 (1) (2010), 471-482.

[7] S. Pehlivan, D. Han and R. Mohapatra, Linearly connected sequences and spectrally optimal dual framesfor erasures, J. Functional Anal. 265 (11) (2013), 2855-2876.

42

Johnson pseudo-contractibility of certain Banach algebras and theirnilpotent ideals

M. Askari-sayah*, A. Pourabbas (Amirkabir University of Technology, Tehran, Iran)A. Sahami( Ilam University, Ilam, Iran)

Abstract: In this paper, we study the notion of Johnson pseudo-contractibility for certain Banachalgebras. For a bicyclic semigroup S, we show that `1(S) is not Johnson pseudo-contractible.We give a necessary and sufficient condition for a Fourier algebra to be Johnson pseudo-contractible. Alsofor an SIN -group G we show that Johnson pseudo-contractibility of S1(G) is equivalent with amenabilityof G.

keywords. Johnson pseudo-contractibility, Semigroup algebras, Segal algebras, Fourier algebras

subject. 43A20, 20M18, 43A20.

1 Introduction

The theory of amenability for Banach algebras was developed by Johnson [5] in 1972. This theory hasbeen occupying an important place in research in modern analysis. But amenability for Banach algebrasremains a too strong concept. This fact leads some theorists to introduce other concepts by relaxing somedifferent conditions in the definition of amenable Banach algebras. F. Ghahramani and R. Loy [3] defined theconcept of approximate amenability for Banach algebras. Pseudo-amenability and pseudo-contractibility,two important generalizations of amenability for Banach algebras, were introduced by F. Ghahramani andY. Zhang in [4]. These notions have been studied for various classes of Banach algebras we may mention,for example, semigroup algebras, Segal algebras and Fourier algebras.

Recently, the second and third named authors defined a new concept of amenability called Johnsonpseudo-contractibility [7]. Johnson pseudo-contractibility of some certain Banach algebras like group alge-bras, measure algebras and Lipschitz algebras were studied in [7]. Furthermore, the second and third authorsin [7, Example 4.1-(iii)] showed that MN(C) (The Banach algebra of all N×N-matrices (zij) with entries inC such that ‖(zij)‖ =

∑i,j|zij | <∞, `1-norm and matrix multiplication) is not Johnson pseudo-contractible

but MN(C) is pseudo-amenable.The purpose of this paper is to study Johnson pseudo-contractibility of certain Banach algebras including

semigroup algebras, Segal algebras and Fourier-Stieltjes algebras. We show that for a unital Banach algebraA, Johnson pseudo-contractibility is equivalent with amenability. This allows us to characterize Johnsonpseudo-contractibility of Fourier-Stieltjes algebras and Johnson pseudo-contractibility of a semigroup algebraprovided that the semigroup has an identity. As an application we show that for N with maximum as itsproduct, the semigroup algebra is not Johnson pseudo-contractible (but it is pseudo-amenable). Againusing this criteria we can see that `1(S) fails to be Johnson pseudo-contractible, whenever S is a bicyclicsemigroup. Also we show that in particular cases Johnson pseudo-contractibility of S1(G) is equivalent withamenability of G.

43

M. Askari-sayah, A. Pourabbas and A. Sahami

The existence of non-zero nilpotent ideals in amenable Banach algebras have been investigated by manyauthors. Firstly, In 1989, P. Curtis and R. Loy showed that non-zero nilpotent ideals in amenable Banachalgebras must be infinite dimensional. Also, nilpotent ideals which have approximation property in amenableBanach algebras was studied by R. Loy and G. Willis in 1994. Inspired by these works, Y. Zhang definedthe concept of approximately complemented subspaces of Banach spaces. Moreover, he showed that apseudo-contractible Banach algebra does not have a non-trivial closed nilpotent ideal which is approximatelycomplemented. The second purpose of the paper is to study nilpotent ideals in Johnson pseudo-contractibleBanach algebras. We show that a Johnson pseudo-contractible Banach algebra A has no non-zero closednilpotent ideal which is complemented in A.

We recall some standard notation and definitions. Suppose that X and Y are normed spaces. Theprojective tensor product of X and Y is denoted by X ⊗p Y . For a Banach algebra A the product map onA determines a map πA : A ⊗p A → A, specified by πA(a ⊗ b) = ab for all a, b ∈ A. The projective tensorproduct A⊗p A becomes a Banach A-bimodule with the following module actions:

a · (b⊗ c) = ab⊗ c, (b⊗ c) · a = b⊗ ca (a, b, c ∈ A),

and with these actions πA becomes an A-bimodule morphism. We regard the dual space A∗ as a BanachA-module with the operations defined by (af)(b) = f(ba), (fa)(b) = f(ab) for all a, b ∈ A and f ∈ A∗. Thefirst Arens product on the the second dual A∗∗ of A is defined by (FH)(f) = F (Hf), where (Hf)(a) =H(fa) for all F,H ∈ A∗∗, f ∈ A∗ and a ∈ A. We denote the unitization of A by A].

Suppose that A is a Banach algebra. A is said to be amenable if it has a virtual diagonal, that is,there exists an element M ∈ (A ⊗p A)∗∗ such that a ·M = M · a and π∗∗A (M)a = a for every a ∈ A. A ispseudo-amenable (pseudo-contractible) if there exists a net (mα) in A⊗p A such that a ·mα −mα · a→ 0(a · mα = mα · a) and πA(mα)a − a → 0, for every a ∈ A, respectively. Moreover, A is called Johnsonpseudo-contractible if there exists a net (mα) in (A⊗pA)∗∗ such that a ·mα = mα ·a and π∗∗A (mα)a−a→ 0,for every a ∈ A. Note that Johnson pseudo-contractibility is strictly weaker than pseudo-contractibility, see[7, Example 4.1-(ii)]. A Banach algebra A is called approximately amenable if there are nets (Mα) ⊆ A⊗pA,(Fα) ⊆ A and (Gα) ⊆ A such that for every a ∈ A

(i) a ·Mα −Mα · a+ Fα ⊗ a− a⊗Gα → 0,

(ii) aFα → a, Gαa→ a, and

(iii) πA(Mα)a− Fαa−Gαa→ 0.

There are some equivalent definitions of amenable and approximately amenable Banach algebras, see [5] and[3].

2 Johnson pseudo-contractibility of certain semigroup algebras

theorem 2.1. Let A be a Banach algebra with an identity. Then the following are equivalent

i) A is Johnson pseudo-contractible,

ii) A] is Johnson pseudo-contractible,

iii) A is amenable.

remark 2.2. It is well-known that a Banach algebra A is amenable if and only if A] is amenable. Thisproperty is not valid in Johnson pseudo-contractibility case. To see this let S = N∪0. With the followingaction

m ∗ n =

m if m = n

0 if m 6= n,

44

Johnson pseudo-contractibility of certain Banach algebras and their nilpotent ideals

S becomes a semigroup. `1(S) is pseudo-contractible. Thus `1(S) is Johnson pseudo-contractible. Weclaim that `1(S)] is not Johnson pseudo-contractible. Suppose conversely that `1(S)] is Johnson pseudo-contractible. Then by Theorem 2.1, `1(S)] is amenable. So `1(S) is amenable and [2, Proposition 10.5 (ii)]implies that the set of idempotents of S is finite, a contradiction.

example 2.3. Let S be a bicyclic semigroup, that is, a semigroup generated by two elements p and q suchthat pq = e 6= qp, where e is the identity for S. It is well-known that `1(S) is not amenable. But since `1(S)is unital, Theorem 2.1 implies that `1(S) is not Johnson pseudo-contractible.

A semigroup S is called inverse if for each s ∈ S there exists a unique element t ∈ S such that sts = sand tst = t. The set of idempotents of semigroup S is denoted by ES .

corollary 2.4. Let S be an inverse semigroup with unit. Then `1(S) is Johnson-pseudo-contractible if andonly if ES is finite and each maximal subgroup of S is amenable.

corollary 2.5. Let S be a semigroup with unit. Then the Banach algebra `1(S) is Johnson-pseudo-contractible if and only if the minimum ideal K(S) exists, K(S) is an amenable group, and S has a principalseries

S = S1 ) S2 ) · · · ) Sm−1 ) Sm = K(S)

such that each quotient Si\Si+1 is a regular Rees matrix semigroup of the form M0(G,P, n), where n ∈ N,G is an amenable group, and the sandwich matrix P is invertible in Mn(`1(G)).

theorem 2.6. Let A be a Banach algebra and let A∗∗ be Johnson pseudo-contractible. Then A is Johnsonpseudo-contractible.

example 2.7. Consider the semigroup N∨ with the semigroup multiplication m ∨ n = maxm,n for eachm,n ∈ N. We claim that `1(N∨) is not Johnson pseudo-contractible. Since `1(N∨) is unital, if `1(N∨) isJohnson pseudo-contractible, then by Corollary 2.4 EN∨ is finite which is a contradiction. Furthermore weknow that `1(N∨) is approximately amenable. Since `1(N∨) is unital, [4, Proposition 3.2] implies that `1(N∨)is pseudo-amenable. Therefore we have a Banach algebra among semigroup algebras which is not Johnsonpseudo-contractible but it is pseudo-amenable and approximately amenable. Moreover, by Theorem 44`1(N∨)∗∗ is not Johnson pseudo-contractible.

definition 2.8. The FourierStieltjes algebra B(G) of G consists of all functions

G −→ C, x 7−→ 〈π(x)ξ, η〉,

where π is some continuous unitary representation of G on a Hilbert space H and ξ, η ∈ B(H).

corollary 2.9. The following are equivalent for a locally compact group G:

(i) B(G) is Johnson pseudo-contractible.

(ii) G has a compact, Abelian subgroup of finite index.

Let G be a locally compact group. A linear subspace S1(G) of L1(G) is said to be a Segal algebra on Gif it satisfies the following conditions

(i) S1(G) is dense in L1(G),

(ii) S1(G) with a norm || · ||S1(G) is a Banach space and ||f ||L1(G) ≤ ||f ||S1(G) for every f ∈ S1(G),

(iii) S1(G) is left translation invariant (i.e. Lyf ∈ S1(G) for every f ∈ S1(G) and y ∈ G) and the mapy 7→ Lyf from G into S1(G) is continuous, where Lyf(x) = f(y−1x),

(iv) ||Lyf ||S1(G) = ||f ||S1(G) for every f ∈ S1(G) and y ∈ G.

The group G is said to have small invariant neighborhoods, denoted by SIN -group if in every neighborhoodof the identity there exists a compact invariant neighborhood of the identity.

45

M. Askari-sayah, A. Pourabbas and A. Sahami

theorem 2.10. Let G be an SIN -group. Then the following statements are equivalent:

(i) G is an amenable group,

(ii) S1(G) is Johnson pseudo-contractible,

(iii) S1(G) is pseudo-amenable.

3 Johnson pseudo-contractibility and nilpotent ideals

A subspace E of a normed space X is called complemented in X, if there exists a bounded operatorP : X → E such that P (x) = x for every x ∈ E.

lemma 3.1. Suppose that X is a normed space and E is a complemented subspace of X. Let I : E −→ Xbe the inclusion map. Then the map IdY ⊗ I : Y ⊗p E −→ Y ⊗p X has closed range for any normed spaceY , where IdY is the identity map of Y .

theorem 3.2. Suppose that A is a Johnson pseudo-contractible Banach algebra and N is a closed comple-mented ideal of A and also E is a closed ideal of A satisfying E ⊆ N and EN = 0. If M is any subset ofA with ME ⊆ EA (EM ⊆ AE), then ME = 0 (EM = 0), respectively.

theorem 3.3. Let A be a Johnson pseudo-contractible Banach algebra. Then A has no non-zero closednilpotent ideal which is complemented in A.

Since every finite dimensional subspace of a Banach algebra is complemented we have the followingcorollary:

corollary 3.4. Every Johnson pseudo-contractible Banach algebra has no non-zero finite dimensional nilpo-tent ideal.

It is well-known that every closed subspace of a Hilbert space is complemented, and up to isomorphismthe only Banach spaces with this property are Hilbert spaces.

Also in a commutative Banach algebra, the ideal generated by a nilpotent element is nilpotent. So wehave the following corollary:

corollary 3.5. Let A be a Johnson pseudo-contractible Banach algebra whose underlying space is a Hilbertspace. Then A has no non-zero nilpotent closed ideal. Moreover if A is commutative, then A has nonon-trivial nilpotent element.

References

[1] M. Askari-sayah, A. Sahami, A. Pourabbas, Johnson pseudo-contractibility of certain Banach algebrasand their nilpotent ideals, Anal. Math. to appear.

[2] H. G. Dales, A. T.-M. Lau, and D. Strauss, Banach algebras on semigroups and on their compactifica-tions, Mem. Amer. Math. Soc. 205 (2010), no. 966, vi+165. MR 2650729

[3] F. Ghahramani and R. J. Loy, Generalized notions of amenability, J. Funct. Anal. 208 (2004), no. 1,229–260. MR 2034298

[4] F. Ghahramani and Y. Zhang, Pseudo-amenable and pseudo-contractible Banach algebras, Math. Proc.Cambridge Philos. Soc. 142 (2007), no. 1, 111–123. MR 2296395

[5] B. Johnson, Cohomology in Banach algebras, American Mathematical Society, Providence, R.I., 1972,Memoirs of the American Mathematical Society, No. 127. MR 0374934

[6] A. Sahami, A. Pourabbas, Johnson pseudo-contractibility of certain semigroup algebras, SemigroupForum 97 (2018), no. 2, 203–213. MR 3852768

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Johnson pseudo-contractibility of certain Banach algebras and their nilpotent ideals

[7] A. Sahami, A. Pourabbas, Johnson pseudo-contractibility of various classes of Banach algebras, Bull.Belg. Math. Soc. Simon Stevin 25 (2018), no. 2, 171–182. MR 3819120

47

Some results on woven frames in Banach Spaces

S. Basati* (University of Maragheh, Maragheh, Iran)

Abstract: In this paper, we generalize the notion of Operetor Banach Frame for woven-weaving framesin Banach frame. Also, some relations between various types of woven-weaving frames in Banach frames arediscussed. We improve some properties and concepts of woven-weaving frames for Banach frames.

keywords. Frame, Woven Frame, Operator Banach Frame, Controlled g-woven


1 Introduction

Frames for Hilbert space were first introduced by Duffin and Schaeffer [5] to study non-harmonic Fourierseries and popularized in 1952. After some decades, Daubechies, Grossmann and Meyer reintroduced frameswith extensive studies, in 1986 [5]. Frames are basis-like building blocks that span a vector space but allowfor linear dependency, which is useful to reduce noise, find amesparse representations, spherical codes, com-pressed sensing, signal processing, wavelet analysis etc., see [3]. Recently, Bemrose, Casazza et.al. in [2, 2]proposed weaving frames in a separable Hilbert space. Weaving frames have potential applications in wire-less sensor networks that require distributed processing under different frames in Hilbert spaces. Weightedand controlled frames have been recently to improve the numerical efficiency of iteractive algorithms forinverting the frame operator.Frames were extended to Banach spaces by Feichtinger and Grochenig [6] who introduced the notion ofatomic decompositions for Banach spaces. Later, Grochenig [7] introduced a more general concept calledBanach frame for Banach Spaces.We are using the phrases ”Operator Banach frame and Operator Banach woven” briefly of the symbols OBFand OBW, respectively. Shekhar in [9] introduced the definition of Operator Banach frame (OBF) and gavesufficient and necessary conditions, for Operator Bessel sequence to be OBF. We present sufficient conditionfor the finite sum of OBWs to be an OBW.Throughout the paper H denotes a separable, infinite-dimensional Hilbert space, [m] = 1, 2, ...,m , X, Y, Zare separable Banach spaces with dual X∗, Y ∗, Z∗ and I denotes the indexing set where as finite or infinitycountable. L (H,K) and L (H) is the collection of all bounded linear operators from H into K and H re-spectively. Let GL(H) be the set of all bounded operators with a bounded inverse and GL+(H) be the setof positive operators in GL(H).

definition 1.1. A family of vectors Φ = ϕii∈I in H is said to be a frame if there are constants 0 < A ≤B <∞ so that for all x ∈ H,

A‖x‖2 ≤∑

i∈I| 〈x, ϕi〉 |2 ≤ B‖x‖2,

where A and B are lower and upper frame bounds, respectively. If only B is assumed, then it is calledB-Bessel sequence. If A = B, it is said A-tight frame and if A = B = 1, it is called a Parseval frame.

48


If Φ = ϕii∈I is a Bessel sequence for H, then the synthesis operator of Φ is the operator

T : l2(I)→ H, T ci :=∑

i∈Iciϕi,

and the adjoint of T is the analysis operator

T ∗ : H → l2(I), T ∗x := 〈x, ϕi〉i∈I.The frame operator S : H → H is defined by S := TT ∗,

Sx = TT ∗x =∑

i∈I〈x, ϕi〉ϕi, ∀x ∈ H.

For the frame ϕi , the operator S is positive, self-adjoint, invertible and AI ≤ S ≤ BI.

definition 1.2. A family of frames fiji∈I,j∈[m] for a Hilbert space H is said to be a woven frame if thereexist universal constants A and B so that for every partition σjj∈[m] of I, the family fiji∈σj ,j∈[m] isa frame for H with lower and upper frame bounds A and B, respectively. For every j ∈ [m], the framesfiji∈σj are called weaving frame.

definition 1.3. Let X be a Banach space and let Xd be an associated Banach space of scalar valuedsequences indexed by N. Let fn ⊂ X∗ and S : Xd → X be given. The pair (fn , S) is called a Banachframe for X with respect to Xd if :

1. fn(x) ∈ Xd, for each x ∈ X.

2. There exist positive constants A and B with 0 < A ≤ B <∞ such that

A ‖x‖X ≤ ‖fn(x)‖Xd ≤ B ‖x‖X , ∀x ∈ X. (11)

3. S is a bounded linear operator such that S(fn(x)) = x, for all x ∈ X.

The positive constants A and B are called the lower and upper frame bounds of the Banach frame (fn , S),respectively. The operator S : Xd → X is called the reconstruction operator ( or the pre-frame operator).The inequality 1.1 is called the frame inequality. The Banach frame (fn , S) is called tight if A = B andnormalized tight if A = B = 1. If removal of one fn renders the collection fn ⊂ X∗ no longer a Banachframe for X, the (fn , S) is called an exact Banach frame.

definition 1.4. A sequence space Xd is called BK-space, if it is a Banach space and the coordinate

functionals ak → ak are continuous on Xd, i.e., the relations xn =α

(n)j

, x = αj ∈ Xd, limn→∞ xn =

x imply

limn→∞

α(n)j = αj (j = 1, 2, ...).

A BK-space is called solid if whenever ak and bk are sequences with bk ∈ Xd and |ak| ≤ |bk| , foreach k ∈ I, then it follows that ak ∈ Xd and

‖ak‖Xd ≤ ‖bk‖Xd .

A sequence space Xd is called an AK-space if it is a topological vector space and

ak = limnρn(ak), ∀ ak ∈ Xd,

where ρn(ak) = (a1, a2, ..., an, 0, ...).

49

S. Basati

If the canonical vectors form a Schauder basis for Xd, then Xd is called a CB-space and its canonicalbasis is denoted by ej∞1 . If Xd is reflexive and a CB-space, then Xd is called an RCB-space. Also, thedual of Xd is denoted by X∗d .

For each family of subspace

(Xd)j

j∈[m]

of l2(I), σj ⊂ I, we have

(Xd)j

j∈[m]

=

α

(n)ij

i∈σj ,j∈[m]

|α(n)ij ∈ C, xn =

α

(n)ij

i∈σj ,j∈[m]

, x = αiji∈σj ,j∈[m]

.

That limn→∞ xn → x also limn→∞ α(n)ij = αij , we define the space:

∑

j∈[m]

⊕(Xd)j

l2

=

α

(n)ij

i∈σj ,j∈[m]

| αij(n) ∈ (Xd)j , ∀j ∈ [m]

,

with the semi - inner product

[α(n)

ij i∈I,j∈[m], α′iji∈I,j∈[m]

]=

∑

i∈I,j∈[m]

|αijα′ij |.

Throughout the paper, when we use the dual X∗d of a BK-Space Xd having the canonical unit vectorsas a basis, we will identify X∗d with its isometrically isomorphic BK-Space constructed by the Lemma 2.6in [3].

proposition 1.5. Suppose that Xd is a BK-space, for which the canonical unit vectors eiji∈I,j∈[m] forms

a Schauder basis. Then fiji∈I,j∈[m] ⊆ X∗ is an X∗d -Bessel woven for X with universal bound B if and onlyif the operator

T : cij −→∑

i∈I,j∈[m]

cijfij

is well defined ( hence bounded ) from Xd into X∗d and ‖T‖ ≤ B.

definition 1.6. Let X be a Banach space , Xi be a sequence of Banach Spaces and Ti ∈ B(X;Xi); i ∈ I.Let A be an associated Banach and S : A → X be an operator . Then (Ti , S) is called an OperatorBanach frame(OBF) for E with respect to A if

1. Tif ∈ A, f ∈ X.

2. there exist constants A and B with 0 < A ≤ B <∞ such that

A ‖f‖X ≤ ‖Tif|A ≤ B ‖f‖X , f ∈ X.

3. S is a bounded linear operator such that

S (Tif) = f, f ∈ X.

The positive constants A and B, respectively, are called lower and upper frame bounds for the OBF (Ti , S).The inequality (2) is called the frame inequality for the OBF. The operator S : A → E is called thereconstruction operator. If condition (1) and the upper inequality in (2) is satisfied, then we call Ti to bean operator Bessel sequence for X with respect to A. Let us denote by Bess(X) the set of all Operator Besselsequences for X with respect to A. For a sequence Ti ∈ Bess(X) define RT : X → A by RT f = Tif ,for all f ∈ X. Then RT ∈ B (X,A) . We call RT the analysis operator for the operator Bessel sequence Ti .

In the following result, we construct a normalized OBW using a given combination OBWs.

50


theorem 1.7. Let(fiji∈I,j∈[m] , S

),(giji∈I,j∈[m] , T

)be OBW for X, Y with respect to A,B respec-

tively. Let Riji∈I,j∈[m] ∈ B(X,Xi), liji∈I,j∈[m] ∈ B(Y, Yi) such that fij +Rij + gij + liji∈I,j∈[m] is

Operator Bessel wovens for X ∪ Y with respect to A∪B and with bound k ≤ ‖S‖−1 + ‖T‖−1 . Then, there

exist a reconstruction operator U : A∪B → X ∪Y such that(Rij + liji∈I,j∈[m] , U

), is a normalized tight

OBW for E ∪ F with respect to A ∪ B.

definition 1.8. Let C,C ′ ∈ GL+(H), the family Λ = Λi ∈ L(H,Hi) : i ∈ I will be called a (C,C ′)-controlled g-frame for H, if Λ is a g-Bessel sequence and there exists constants A > 0 and B < ∞ suchthat

A ‖f‖2 ≤∑

i∈I

⟨ΛiCf,ΛiC

′f⟩≤ B ‖f‖2 , ∀f ∈ H.

A and B will be called controlled frame bounds. If C ′ = I, we call Λ = Λii∈I a C-controlled g-frame for Hwith bounds A and B. If the second part of the above inequality holds, it will be called a (C,C ′)-controlledg-Bessel sequence with bound B.

definition 1.9. Let C,C ′ ∈ GL+ (H) . The family

Λ = Λij ∈ L (H,Hij) : i ∈ I, j ∈ [m]

will be called a (C,C ′)-controlled g-woven for H, if Λ is a g-Bessel sequence and there exists constants A > 0and B <∞ such that

A ‖f‖2 ≤∑

i∈I,j∈[m]

⟨ΛijCf,ΛijC

′f⟩≤ B ‖f‖2 , ∀f ∈ H.

A and B will be called controlled frame bounds. If C ′ = I, we call Λ = Λiji∈I,j∈[m] a C-controlledg-woven for H with bounds A and B. If the second part of the above inequality holds, it will be called a(C,C ′)-controlled g-Bessel sequence with bound B.

proposition 1.10. Let C,C ′ ∈ GL+(H). The family

Λ = Λij ∈ L (H,Hij) : i ∈ I, j ∈ [m]

andΘ =

Θij ∈ L

(H,Hij

): i ∈ I, j ∈ [m]

are g-wovens if and only if Λ + Θ is a C2-controlled g-woven.

References

[1] T. Bemrose, P . G. Casazza, K. Grochenig, M. C. Lammers and R.G. Lynch, Weaving Frames, Oper.Matrices, 10 (4) (2016), 1093-1116.

[2] P. G. Casazza, R. G. Lynch, Weaving properties of Hilbert space frames, J. Proc. SampTA. (2015),110-114.

[3] P. G. Casazza and G. Kutyniok, Finite Frames: Theory and Applications, Birkhauser (2012).

[4] P. Casazza, O. Christensen, D. Stoeva, Frame expansions in separable Banach spaces, J. Math. Anal.Appl. 307 (2005) 710-723.

[5] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27,(1986) 12711283.

[6] R. J. Duffine and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72(1952), 341-366.

51

S. Basati

[7] H. G. Feichtinger and K.H. Grochenig, Banach spaces related to integrable group representation andtheir atomic decomposition, I J. Funct. Anal.,86(1989),307-340.

[8] K. Grochenig, Describing functions: atomic decompositions versus frames. Monatsh. Math., 112 No.1(1991), 1-41.

[9] Ch. Shekhar, Operator Banach frames in Banach spaces, 2217-3412.

52

Woven frames in Banach spaces

S. Basati* and A. Rahimi

(University of Maragheh, Maragheh, Iran)

Abstract: Banach frames are defined by straightforward generalization of (Hilbert space) frames. Inthis article, we investigate that perturbed frames is obtained as the image of a bounded, invertible operatorof a given frame. Also, we improve the notion of woven-weaving frames to Banach frames.

keywords. frame, woven frame, Banach frame


1 Introduction

Duffin and Schaeffer [5] introduced frames for Hilbert spaces in the context of nonharmonic Fourier series.Today frames have applications as signal processing, image processing, data compression, and compressedsensing, spherical codes, wavelet analysis etc, for example see [4], [3].Casazza, et. al. in [2] proposed woven-weaving frames in separable Hilbert space. The concept of weavingframes is motivated by a problem regarding distributed signal processing where redundant building blocks( frames ) play an important role. For example, in wireless sensor networks where frames may be subjectedto distributed processing under different frames.Frames were extended to Banach spaces by Feichtinger and Grochenig [6] who introduced the notion ofatomic decompositions for Banach spaces. Later, Grochenig [7] introduced a more general concept calledBanach frame for Banach Spaces. Balan [1] introduced the concept of a vector-valued frame (or superframe).The vector-valued frame has significant applications in mobile communications, staellite communications,and computer area networks. The concept of vector-valued (super) weaving frame studied by Deepshikhaand Vashisht in [8].Throughout the paper, H denotes a separable, infinite-dimensional Hilbert space, [m] = 1, 2, ...,m, X,Y, Zare separable Banach spaces with dual X∗, Y ∗, Z∗ and I denotes the indexing set where as finite or infinitycountable. We take Fj ⊂ Hj for j ∈ [L]. If F1, ..., FL, I is vector-valued frame (Bessel sequence) forH1 ⊕ ...⊕Hn, then Fj(j = 1, ..., L) are frames (Bessel sequence) for atomic spaces Hj .

definition 1.1. A family of vectors Φ = ϕii∈I in H is said to be a frame if there are constants 0 < A ≤B <∞ so that for all x ∈ H,

A‖x‖2 ≤∑

i∈I| 〈x, ϕi〉 |2 ≤ B‖x‖2,

where A and B are lower and upper frame bounds, respectively. If only B is assumed, then it is calledB-Bessel sequence. If A = B, it is said A-tight frame and if A = B = 1, it is called a Parseval frame.

53

S. Basati, A. Rahimi

If Φ = ϕii∈I is a Bessel sequence for H, then the synthesis operator of Φ is the operator

T : l2(I)→ H, T ci :=∑

i∈Iciϕi,

and the adjoint of T is the analysis operator

T ∗ : H → l2(I), T ∗x := 〈x, ϕi〉i∈I.

The frame operator S : H → H is defined by S := TT ∗,

Sx = TT ∗x =∑

i∈I〈x, ϕi〉ϕi, ∀x ∈ H.

For the frame ϕi , the operator S is positive, self-adjoint, invertible and AI ≤ S ≤ BI.

definition 1.2. A family of frames fiji∈I,j∈[m] for a Hilbert space H is said to be a woven frame if thereexist universal constants A and B so that for every partition σjj∈[m] of I, the family fiji∈σj ,j∈[m] isa frame for H with lower and upper frame bounds A and B, respectively. For every j ∈ [m], the framesfiji∈σj are called weaving frame.

example 1.3. There exist two Parseval frames that give weaving with arbitrary weaving bounds. Let ε > 0,

set δ = (1 + ε2)−12 , and let e1, e2, ...en be the standard orthonormal basis of Rn, then the sets

φ = ϕini=1 = δe1, δεe1, δe2, δεe2, ..., δen, δεen

and

ψ = ψini=1 = δεe1, δe1, δεe2, δe2, ..., δεen, δen

are Parsevsal frames, which are woven since any choice of σ gives a spanning set.

definition 1.4. Let I be a countable index set and consider

(F1;π1; I), ..., (FL;πL; I) , L

indexed sets of vectors ( not necessarily from the same Hilbert space ), where πk : I→ Fk is the correspondingindexing map. A collection of such countable sets of vectors toghether with their corresponding indexingmaps from a same index set is called a superset. In short, we write (F1, ..., FL) for a superset when anindexing by a same index set I for each subset Fk of vectors of some Hilbert space Hk ( or a bigger spaceKk ) is fixed. We write

F = F1 ⊕ ...⊕ FL =f1i ⊕ ...⊕ fLi : i ∈ I

, fki = πk(i) ∈ Fk.

Note that the space H1 ⊕H2 ⊕ ...⊕HL is a Hilbert space with natural inner product

〈f1 ⊕ ...⊕ fL, g1 ⊕ ...⊕ gL〉 =L∑

i=1

〈fi, gi〉Hi , fi, gi ∈ Hi (1 ≤ i ≤ L) .

definition 1.5. The superset (F1, ..., FL) is called a vector-value frame or super frame if F is a frame forthe space H1 ⊕ H2 ⊕ ... ⊕ HL. That is, if there exist finite positive numbers A0 ≤ B0 such that for everyhk ∈ Hk (1 ≤ k ≤ L) ,

A0

(‖h1‖2 + ...+ ‖hL‖2

)≤∑

i∈I|L∑

k=1

⟨hk, f

ki

⟩|2 ≤ B0

(‖h1‖2 + ...+ ‖hL‖2

).

54

Woven frames in Banach spaces

definition 1.6. A family of vector-valued framesF i1, ..., F

iL, I

: i ∈ [m]

for H1 ⊕H2 ⊕ ... ⊕HL is saidto be woven if there exist universal constants A,B such that for any partition σii∈[m] of I, the family

⋃

i∈[m]

F i1, ..., F

iL, σi

is a vector-valued frame for H1 ⊕H2 ⊕ ...⊕HL with lower and upper frame bounds A and B, respectively.For every i ∈ [m], the vector-valued frames

⋃i∈[m]

F i1, ..., F

iL, σi

are called vector-valued weavings ( or

simply weaving ).

example 1.7. Let L = m = 3; I = N and let H1 = H2 = H3 = l2(N), define F1 = f1ii∈I , G1 = g1ii∈I ,L1 = l1ii∈I ⊂ H1, as follows :

f1i =

ej , i = 23j − 5

0, otherwise, g1i =

ej , i = 23j − 3

0, otherwiseand l1i =

ej , i = 23j − 5, 23j − 3

0, otherwise

and F2 = f2ii∈I, G2 = g2ii∈I, L2 = l2ii∈I ⊂ H2, as follow:

f2i =

ej , i = 23j

0, otherwise, g2i =

ej , i = 23j − 4


ej , i = 23j, 23j − 2

0, otherwise

and F3 = f3ii∈I, G3 = g3ii∈I, L3 = l3ii∈I ⊂ H3, as follows:

f3i =

ej , i = 23j − 1

0, otherwise, g3i =

ej , i = 23j − 3


ej , i = 23j − 1, 23j − 5

0, otherwise

where eii∈I is the canonical orthonormal basis of l2 (N) . We will prove that for every partision σ ⊂ I thevector-valued families F1, F2, F3, I, G1, G2, G3, I and L1, L2, L3, I forming woven-weaving frames.

definition 1.8. A sequence space Xd is called BK-space, if it is a Banach space and the coordinate

functionals ak → ak are continuous on Xd, i.e., the relations xn =α

(n)j

, x = αj ∈ Xd, limn→∞ xn =

x imply

limn→∞

α(n)j = αj (j = 1, 2, ...).

A BK-space is called solid if whenever ak and bk are sequences with bk ∈ Xd and |ak| ≤ |bk| , foreach k ∈ I, then it follows that ak ∈ Xd and

‖ak‖Xd ≤ ‖bk‖Xd .

A sequence space Xd is called an AK-space if it is a topological vector space and

ak = limnρn(ak), ∀ ak ∈ Xd,

where ρn(ak) = (a1, a2, ..., an, 0, ...).

definition 1.9. Let X be a Banach space and let Xd be an associated Banach space of scalar valuedsequences indexed by N. Let fn ⊂ X∗ and S : Xd → X be given. The pair (fn , S) is called a Banachframe for X with respect to Xd if :

1. fn(x) ∈ Xd, for each x ∈ X.

2. There exist positive constants A and B with 0 < A ≤ B <∞ such that

A ‖x‖X ≤ ‖fn(x)‖Xd ≤ B ‖x‖X , ∀x ∈ X. (12)

55

S. Basati, A. Rahimi

3. S is a bounded linear operator such that S(fn(x)) = x, for all x ∈ X.

The positive constants A and B are called the lower and upper frame bounds of the Banach frame (fn , S),respectively. The operator S : Xd → X is called the reconstruction operator ( or the pre-frame operator).The inequality 1.1 is called the frame inequality. The Banach frame (fn , S) is called tight if A = B andnormalized tight if A = B = 1. If removal of one fn renders the collection fn ⊂ X∗ no longer a Banachframe for X, the (fn , S) is called an exact Banach frame.

theorem 1.10. Let (fn , S) (where fn ⊂ X∗, S : Xd −→ X) , (gn , T )(where gn ⊂ Y ∗, T : Yd −→ Y ) and (hn ,K) (where hn ⊂ Z∗,K : Zd −→ Z) be Banach frames for Ba-nach spaces X and Y and Z, respectively. Where Xd, Yd and Zd are BK-spaces. Then there exist a sequencePn ⊂ (X × Y × Z)∗, an associated Banach sequence space (X × Y × Z)d and a reconstruction operatorU : (X × Y × Z)d −→ X × Y × Z such that (Pn , U) is a Banach frame for X × Y × Z with respect to(X × Y × Z)d. Furthermore, if (fn , S), (gn , T ) and (hn ,K) are exact, then (Pn , U) is also exact.

proposition 1.11. Let ϕii∈I be a frame with bounds A and B and V be a bounded operator. If ‖Id −V ‖2 ≤ A

B and ‖V − V 2‖2 ≤ AB , then the frames ϕii∈I, V ϕii∈I and V 2ϕii∈I are woven.

References

[1] R. Balan, Multiplexing of signals using superframes. In: SPIE Wavelets Applications, Vol. 4119 of Signaland Image Processing VIII (2000), 118129.

[2] T. Bemrose, P . G. Casazza, K. Grochenig, M. C. Lammers and R.G. Lynch, Weaving Frames, Oper.Matrices, 10 (4) (2016), 1093-1116.

[3] P. Casazza, O. Christensen, D. Stoeva, Frame expansions in separable Banach spaces, J. Math. Anal.Appl. 307 (2005) 710-723.

[4] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, Boston (2016).

[5] R. J. Duffine and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72(1952), 341-366.

[6] H. G. Feichtinger and K.H. Grochenig, Banach spaces related to integrable group representation andtheir atomic decomposition, I J. Funct. Anal.,86(1989),307-340.

[7] K. Grochenig, Describing functions: atomic decompositions versus frames. Monatsh. Math., 112 No.1(1991), 1-41.

[8] L. K. Vashisht, Deepshikha, Vector-valued (super) weaving frames, 20 Apr 2018.

56

Vector-valued Φ-contractibility of Banach algebras

M. Dashti* (Malayer University, Malayer, Iran)S. Soltani Renani (Isfahan University of Technology, Isfahan, Iran)

Abstract: Let A be a Banach algebra and let M be a unital Banach algebra. For a homomorphism Φfrom A into M, we introduce the concept of vector-valued Φ-contractibility which generalize the notion ofφ-contractibility. We also give characterizations of this concept in terms of projectivity.

keywords. Banach algebra, Banach module, homological property

subject. 43A07, 46H05, 46L10

1 Introduction and preliminaries

The concepts of injectivity and projectivity of Banach modules were introduced and studied by A. Ya.Helemski [4]. Helemski obtained a characterization of amenability of Banach algebras by homological prop-erties: a Banach algebra A is amenable if and only if the first cohomology group, H1(A,X ∗) vanishes forevery Banach A-bimodule X .

Given a multiplicative linear functional φ on a Banach algebra A, the term φ-amenability was coinedindependently by several authors; see [5] and [6]. Recently, the notion of vector-valued invariant mean onspaces of bounded linear maps was introduced and studied by the authors and R. Nasr-Isfahani [2], whichin the special case is equivalent to the φ-amenability.

The concept (left) φ-contractibility was introduced and studied by Hu, Monfared and Traynor [5]. ABanach algebra A is (Left) φ-contractible if H1(A,X ) = 0 for all Banach A-module X with right actionx · a = φ(a)x (a ∈ A, x ∈ X ). This concept was introduced and studied by Hu, Monfared and Traynor [5].Later on, the second author and R. Nasr-Isfahani [7] provided alternative characterization of φ-contractibilityin terms of homological properties. They also proved that φ-contractibility is equivalent to the existence ofan element m ∈ A such that φ(m) = 1 and am = φ(a)m for all a ∈ A.

Let M be a unital Banach algebra. For a homomorphism Φ from A into M, we introduce the con-cept of vector-valued Φ-contractibility which generalize the notion of φ-contractibility. We also establishcharacterizations of this consept in terms of projectivity.

2 Main results

Let A be a Banach algebra and let M be a unital Banach algebra. Let also Φ be a homomorphism fromA into M. In the sequel, we introduce the concept of vector-valued Φ-contractibility. We start with a fewdefinitions.

Let E and F be two Banach spaces, and let B(E ,F) be the Banach space of all bounded operators fromE into F . In the case where E and F are Banach right A-modules, BA(E ,F) denotes the closed linearsubspace of B(E ,F) consisting of all right A-module morphisms.

We denote by ∆(A,M) the set of all bounded nonzero homomorphisms from A into M. Let us denoteby

57

M. Dashti, S. Soltani Renani

∆u(A,M) = Φ ∈ ∆(A,M), u ∈ Im(Φ).

Let Φ ∈ ∆(A,M). All over this paper, we consider M as a Banach right A-module under following action

ω · a = ωΦ(a) (a ∈ A, ω ∈M).

definition 2.1. Let A be a Banach algebra, let M be a unital Banach algebra with identity element uand let Φ ∈ ∆(A,M). We say that A is vector-valued Φ-contractible if there exists Ψ ∈ BA(M,A) withΦ Ψ = IM.

proposition 2.2. Let A be a unital Banach algebra, let M be a Banach algebra with identity element uand let Φ ∈ ∆(A,M) be an epimorphism. Then the following statements are equivalent.

(i) ker(Φ) has a left identity.

(ii) A is vector-valued Φ-contractible.

Proof. Let e be an identity for A. Since Φ is an epimorphism, it follows that Φ(e) = u.(i)⇒ (ii). Suppose that ker(Φ) has a left identity ι. We define operator Ψ :M→A by

Ψ(ω) = eaω − ιaω (ω ∈M),

where aω ∈ A with Φ(aω) = ω. Let us note that this function is well defined; in fact if Φ(aω) = Φ(aω) = ω,then aω − aω ∈ ker(Φ). So,

ι(aω − aω) = (aω − aω) = e(aω − aω).

This means that eaω − ιaω = eaω − ιaω. Furthermore,

ω · a = ωΦ(a) = Φ(aω)Φ(a) = Φ(aωa) (a ∈ A).

So, Ψ(ω) · a = (e− ι)aωa = Ψ(ω · a) and

Φ(Ψ(ω)) = Φ(eaω − ιaω) = Φ(e)Φ(aω) = ω.

Therefore, A is vector-valued Φ-contractible.(ii)⇒ (i). Assume that A is vector-valued Φ-contractible. Then there exists Ψ ∈ B(M,A) such that

Φ Ψ = IM and Ψ(ω · a) = Ψ(ω) · a,for all a ∈ A. Take ι = e−Ψ(u). Clearly ι ∈ ker(Φ) and for each b ∈ ker(Φ) we have

(e−Ψ(u))b = eb−Ψ(u · b) = eb−Ψ(uΦ(b)) = eb = b.

This means that ι is a left identity of ker(Φ).

The following theorem was obtained in [5, Theorem 6.4].

theorem 2.3. Let A be a Banach algebra and φ ∈ ∆(A,C). Then A is (left) φ-contractible and has a leftidentity if and only if kerφ has a left identity.

For φ ∈ ∆(A,C), we define φu ∈ ∆(A,M) by (φu)(a) = 〈φ, a〉u for all a ∈ A. In the next result, wegive a description of φ-contractiblity of A in terms of vector-valued φu-contractiblity. This result followsfrom Proposition 2.2 and Theorem 2.3.

proposition 2.4. Let A be a unital Banach algebra and let M be a Banach algebra with identity elementu. Let also φ ∈ ∆(A,C). Then A is φ-contractible if and only if A is vector-valued φu-contractible.

The Banach right A-module X is called essential if the linear span of X · A is dense in X . Note that ifΦ ∈ ∆(A,M) with u ∈ Im(Φ) then the Banach right A-module M is essential.

Set P = X⊗A. Then P is a Banach right A-module with the following action

58

Vector-valued Φ-contractibility of Banach algebras

(ξ ⊗ b) · a = ξ ⊗ ba (a, b ∈ A, ξ ∈ X ).

Consider the canonical morphism π ∈ BA(X⊗A,X ) defined by π(ξ ⊗ a) = ξ · a for all a ∈ A and ξ ∈ X .An operator T ∈ B(E ,F) is called admissible if T S T = T for some S ∈ B(F , E). A Banach right

A-module P is called projective if for any Banach right A-module E and F , each admissible epimorphismT ∈ BA(E ,F), and each S ∈ BA(P,F), there exists R ∈ BA(P, E) such that T R = S; that is the followingdiagram commutes:

P

F?

S

E -T

R

The following proposition is proved in [4, Proposition IV.1.1].

proposition 2.5. Let A be a Banach algebra and let X be an essential Banach right A-module. Then Xis projective if and only if the canonical morphism π ∈ BA(X⊗A,X ) is a retraction.

The next result characterizes projectivity of certain Banach right A-modules in terms of vector-valuedΦ-contractibility of A.

theorem 2.6. Let A be a Banach algebra and let M be a unital Banach algebra. Let also Φ ∈ ∆(A,M)be an admissible epimorphism operator. Then the following assertions are equivalent.

(i) A is vector-valued Φ-contractible.

(ii) The Banach right A-module M is projective.

Proof. (i)⇒ (ii). Suppose that A is vector-valued Φ-contractible. So, there exists Ψ ∈ BA(M,A) withΦ Ψ = IM. We define ρ :M→M⊗A by ρ(ω) = u⊗Ψ(ω). Note that

π ρ(ω) = u ·Ψ(ω) = uΦ(Ψ(ω)) = uω = ω,

for all ω ∈M. Furthermore

ρ(ω · a) = u⊗Ψ(ω · a) = u⊗Ψ(ω) · a= (u⊗Ψ(ω)) · a = ρ(ω) · a.

By Proposition 2.5, since M is an essential Banach right A-module, M is projective.To prove (ii)⇒ (i), consider admissible epimorphism Φ ∈ BA(A,M) and identity operator IM ∈

BA(M,M). Since M is projective, then there exists R ∈ BA(M,A) such that Φ R = IM; that is Ais vector-valued Φ-contractible.

Let φ be a multiplicative linear functional on a Banach algebra A. The complex algebra C could beregarded as a commutative A-module via φ; infact

a · α = α · a = αφ(a) (a ∈ A, α ∈ C),

in this case, we use the notation Cφ.We conclude the paper by an immidiate consequence of Theorem 2.5. This characterization of φ-

contractibility is given in [7, Theorem 4.3].

theorem 2.7. Let A be a Banach algebra and φ ∈ ∆(A). Then the following assertions are equivalent.

(i) A is φ-contractible.

(ii) The Banach right A-module Cφ is projective.

59

M. Dashti, S. Soltani Renani

References

[1] H. G. Dales and M. E. Polyakov, Homological properties of modules over group algebras, Proc. LondonMath. Soc. (2004), 390–426.

[2] M. Dashti, R. Nasr-Isfahani and S. Soltani Renani, Vector-valued invariant means on spaces of boundedlinear maps , Colloq. Math. (2013), 1–13.

[3] M. Dashti and S. Soltani Renani, The retraction of certain Banach right modules associated to a char-acter, Math. Slovaca (2018), to appear.

[4] A. Ya. Helemskii, The homology of Banach and topological algebras, Kluwer, Dordrecht, 1989.

[5] Z. Hu, M. S. Monfared and T. Traynor, On character amenable Banach algebras, Studia Math. 193(2009), 53–78.

[6] E. Kaniuth, A. T. Lau and J. Pym, On φ-amenability of Banach algebras, Math. Proc. Cambridge Philos.Soc. 144 (2008), 85–96.

[7] R. Nasr-Isfahani and S. Soltani Renani, Character contractibility of Banach algebras and homologicalproperties of Banach modules, Studia Math. 202 (2011), 205–225.

[8] P. Ramsden, Homological properties of semigroup algebras, Thesis, University of leeds, 2008.

60

Characterizing g-R-duality in Hilbert spaces

F. Enayati* and M. S. Asgari

(Islamic Azad University, Central Tehran Branch, Tehran, Iran)

Abstract: R-dual of certain sequences in Hilbert spaces were introduced by Casazza, Kutyniok andLammers in 2004. In this paper we introduce a new sequence Θii∈I and use it to characterize the g-R-dual sequences in Hilbert spaces. we investigate the relation between the new introduced sequence Θii∈Ifor the g-R-dual sequence Λii∈I .

keywords. G-orthonormal bases, g-frames, g-Riesz-dual sequence, Riesz-duality.


1 Introduction

Duality principles in Gabor Theory play a fundamental role for analyzing Gabor system. Essentially, theduality principle is a result that allows to check the frame condition for Gabor systems in a conceptuallysimpler way. The concept of Riesz-dual of a frame, firstly was introduced by Casazza, kutyniok and Lammersin [2]. Then, the various concepts of the R-duals were introduced by Stoeva, Christensen in [5, 6] with themotivation to obtain a general version of the duality principle in Gabor analysis. In [7, 8], Wenchang Sunintroduced g-frames in Hilbert spaces. The content of this paper is as follows: In the rest of this sectionwe will briefly recall the necessary parts from g-frames, g-orthonormal bases and g-Riesz bases. For moreinformation we refer to [1, 2, 4, 8, 9]. In Sec.2, we also introduce a new sequence Θii∈I and use it tocharacterize the g-R-dual sequences in Hilbert spaces.

Let H and K be two Hilbert spaces and Vii∈I and Wjj∈I be sequences of close subspaces of H andK, where I is a countable (or finite) index set. Let B(H, Vi) be the collection of all bounded linear operatorsfrom H into Vi. Recall that a sequence Λ = Λi ∈ B(H, Vi) i ∈ I is said to be a g-frame for H with respectto Vii∈I if there are two positive constants 0 < C ≤ D <∞ such that:

C‖f‖2 ≤∑

i∈I‖Λif‖2 ≤ D‖f‖2 , ∀f ∈ H. (13)

The constants C and D are called g-frame bounds. If only the right-hand inequalit of (13) is required, wecall it a g-Bessel sequence. We denote the representation space associated with a g-Bessel sequence Λii∈Ias follows:

(∑

i∈I⊕Vi

)`2

=g′ii∈I | g′i ∈ Vi,

∑

i∈I‖g′i‖2 <∞

.

In order to analyze a signal f ∈ H, i.e., to map it into the representation space, the analysis operator TΛ :H →

(∑i∈I ⊕Vi

)`2

given by TΛf = Λifi∈I is applied. The associated synthesis operator, which provides a

61

F. Enayati, M. S. Asgari

mapping from the representation space to H, is defined to be the adjoint operator T ∗Λ :(∑

i∈I ⊕Vi)`2→ H,

which is given by T ∗Λ(g′ii∈I) =∑

i∈I Λ∗i g′i. By composing TΛ and T ∗Λ we obtain the g-frame operator

SΛ : H → H, SΛf = T ∗ΛTΛf =∑

i∈I Λ∗iΛif , which is a positive, self-adjoint and invertible operator and

C ≤ ‖SΛ‖ ≤ D. The canonical dual g-frame for Λii∈I is defined by Λii∈I where Λi = ΛiS−1Λ which

is also a g-frame for for H with respect to Vii∈I with 1D and 1

C as its lower and upper frame bounds,respectively. Also we have

f =∑

i∈IΛ∗i Λif =

∑

i∈IΛ∗iΛif, ∀f ∈ H.

Moreover, ΛiS− 1

2Λ i∈I is a Parseval g-frame for H with respect Vii∈I .

Since almost all applications require a fnite model for their numerical treatment, we restrict ourselvesto a fnite-dimensional space in the following examples.

example 1.1. Let H = Cn+1 and V1 = V2 = . . . = Vn+1 = Cn. Define

Λ1 =

−1 0 0 . . . 0 01 0 0 . . . 0 0...

......

......

1 0 0 . . . 0 0

, Λ2 =

0 1 0 . . . 0 00 −1 0 . . . 0 0...

......

......

0 1 0 . . . 0 0

,

...

Λn =

0 0 0 . . . 1 00 0 0 . . . 1 0...

......

......

0 0 0 . . . −1 0

, Λn+1 =

0 0 0 . . . 0 10 0 0 . . . 0 1...

......

......

0 0 0 . . . 0 1

.

Then, the set Λ = Λin+1i=1 is a tight g-frame for Cn+1 with respect to Cn. Because for each f =

(z1, z2, . . . , zn+1) ∈ Cn+1, we have

n+1∑

i=1

‖Λif‖2 = n(|z1|2 + |z2|2 + . . .+ |zn+1|2) = n‖f‖2.

definition 1.2. Let Ξi ∈ B(H,Wi)| i ∈ I be a sequence of operators. Then

(i) Ξii∈I is a g-complete set for H with respect to Wii∈I , if H = SpanΞ∗i (Wi)i∈I .

(ii) Ξii∈I is an g-orthonormal system for H with respect to Wii∈I , if ΞiΞ∗j = δijIWj for all i, j ∈ I.

(iii) A g-complete and g-orthonormal system Ξii∈I is called an g-orthonormal basis for H with respectto Wii∈I .

(iv) If Λii∈I is g-complete and there are positive constants A and B such that for any finite subset J ⊂ Iand gj ∈ Vj , j ∈ J ,

A∑

j∈J

∥∥gj∥∥2 ≤

∥∥∥∑

j∈JΛ∗jgj

∥∥∥2≤ B

∑

j∈J

∥∥gj∥∥2,

then we say that Λii∈I is a g-Riesz basis for H with respect to Vjj∈J .

example 1.3. Let H = Cn and V1 = V2 = . . . = Vn = C. For k = 1, 2, ..., n, define

Ξk =1√n

[1 e2πi(k−1)/n . . . e2πi(k−1)(n−1)/n

].

62


Then we have

Ξ∗k =1√n

1

e−2πi(k−1)/n

...

e−2πi(k−1)(n−1)/n

.

It is clear that ‖Ξk‖ = 1 and for any 1 ≤ k, ` ≤ n we have

ΞkΞ∗` =

1

n

[1 e2πi(k−1)/n . . . e2πi(k−1)(n−1)/n

]

1

e−2πi(`−1)/n

...

e−2πi(`−1)(n−1)/n

=1

n

n−1∑

j=0

e2πij(k−`)/n = δk`.

Also for any f = (z1, z2, . . . , zn) ∈ Cn we have

n∑

k=1

∣∣Ξkf∣∣2 =

1

n

n∑

k=1

∣∣∣n∑

j=1

zje2πi(k−1)(j−1)/n

∣∣∣2

=1

n

n∑

k=1

( n∑

j=1

zje2πi(k−1)(j−1)/n

)( n∑

m=1

zme2πi(k−1)(m−1)/n

)

=n∑

j=1

n∑

m=1

zjzm

( 1

n

n∑

k=1

e2πi(k−1)(j−m)/n)

=n∑

j=1

n∑

m=1

zjzmδjm =n∑

j=1

|zj |2 = ‖f‖2.

Therefore Ξ = Ξknk=1 is a g-orthonormal basis for Cn with respect to C.

It is well known that there is a close relation between the concepts of g-Riesz bases and g-frames:

lemma 1.4. ( [9] ). Let Λii∈I be sequence of operators for H with respect to Vii∈I . Then the followingstatements are equivalent:

(i) The sequence Λii∈I is a g-Riesz basis for H with respect to Vii∈I .

(ii) The sequence Λii∈I is a g-basis and a g-frame for H with respect to Vii∈I .

proposition 1.5. ( [1] ). Let Λii∈I be a tight g-frame for H with respect to Vii∈I . Then there exists aHilbert spaceM⊇ H and a g-orthonormal basis Ψii∈I forM with respect to Vii∈I such that Λi = ΨiP,where P is the orthogonal projection from M onto H.

definition 1.6. Let Ξ = Ξii∈I and Ψ = Ψii∈I be g-orthonormal bases for H with respect to Wii∈Iand Vii∈I , respectively. Let Λii∈I be a g-Bessel sequence with respect to Vii∈I . The generalizedRiesz-dual sequence (g-R-dual sequence) of Λii∈I with respect to (Ξ,Ψ) is the sequence Γjj∈I given by

Γj : H →Wj , Γj =∑

i∈IΞjΛ

∗iΨi. (14)

example 1.7. Let H = C3n, and let V1 = V2 = . . . = Vn = C3. Define

Ξ1 =

[1 0 0 . . . 0 0 00 1 0 . . . 0 0 00 0 1 . . . 0 0 0

], . . . ,Ξn =

[0 0 0 . . . 1 0 00 0 0 . . . 0 1 00 0 0 . . . 0 0 1

].

63


A direct calculation shows that ‖Ξk‖2 = 1 and ΞkΞ∗l = δkl for any 1 ≤ k, l ≤ n, we also have

n∑

k=1

‖Ξkf‖2 =n∑

k=1

(|z3k−2|2 + |z3k−1|2 + |z3k|2) = ‖f‖2 , ∀f ∈ zi3ni=1 ∈ C3n

Therefore Ξ = Ξini=1 is an g-orthonormal basis for C3n with respect to C3. Similarly, the sequenceΨ = Ψini=1 defined by

Ψ1 =

[0 0 1 . . . 0 0 00 1 0 . . . 0 0 01 0 0 . . . 0 0 0

], . . . ,Ψn =

[0 0 0 . . . 0 0 10 0 0 . . . 0 1 00 0 0 . . . 1 0 0

].

is also an g-orthonormal basis for C3n with respect to C3. The sequence Λ = Λii∈I defined as follows:

Λ1 =

[1 0 0 . . . 0 0 00 2 0 . . . 0 0 00 0 3 . . . 0 0 0

], . . . ,Λn =

[0 0 0 . . . 2n− 1 0 00 0 0 . . . 0 2n 00 0 0 . . . 0 0 2n+ 1

].

Then, the set Λ = Λini=1 is a g-frame for C3n with respect to C3. The g-R-dual sequence for the sequenceΛ with respect to (Ξ,Ψ) is defined as follows:

Γ1 =

[0 0 1 . . . 0 0 00 2 0 . . . 0 0 03 0 0 . . . 0 0 0

], . . . ,Γn =

[0 0 0 . . . 0 0 2n− 10 0 0 . . . 0 2n 00 0 0 . . . 2n+ 1 0 0

]

2 Main results

In this section, we introduce the new sequence Θii∈I and use it to characterize the g-R-dual sequence inHilbert spaces.

definition 2.1. Let Λ = Λii∈I be a g-Bessel sequence for H with respect to Vii∈I and Γ = Γjj∈I bea g-Riesz sequence for H with respect to Wjj∈I . Let Ξ = Ξii∈I be a g-orthonormal base for H withrespect to Wii∈I . For all i ∈ I, define Θi : H → Vi as follows:

Θi =∑

k∈IΛiΞ

∗kΓk, (15)

where Γkk∈I is the dual g- Riesz sequence of Γjj∈I .

Note that under the above assumptions the sequences Γkk∈I and Ξii∈I are g-Bessel sequences,implying that the infinite series defining Θii∈I is convergent.

example 2.2. Let H = C3n and Vi = Wi = C3. Define

Ξ1 =

[1 0 0 . . . 0 0 00 1 0 . . . 0 0 00 0 1 . . . 0 0 0

], . . . ,Ξn =

[0 0 0 . . . 1 0 00 0 0 . . . 0 1 00 0 0 . . . 0 0 1

].

A direct calculation shows that ‖Ξk‖2 = 1 and ΞkΞ∗l = δkl for any 1 ≤ k, l ≤ n, we also have

n∑

k=1

‖Ξkf‖2 =n∑

k=1

(|z3k−2|2 + |z3k−1|2 + |z3k|2) = ‖f‖2 , ∀f ∈ zi3ni=1 ∈ C3n

Therefore Ξ = Ξini=1 is an g-orthonormal basis for C3n with respect to C3. The sequence Λ = Λini=1

defined as follows:

Λ1 =

[1 0 0 . . . 0 0 00 −1 0 . . . 0 0 00 0 −1 . . . 0 0 0

], . . . ,Λn =

[0 0 0 . . . 1 0 00 0 0 . . . 0 −1 00 0 0 . . . 0 0 −1

]

64


the set Λ = Λini=1 is a g-Bessel for C3n with respect to C3. Similarly, the sequence Γ = Γii∈I defined by

Γ1 =

[1 0 0 . . . 0 0 00 2 0 . . . 0 0 00 0 3 . . . 0 0 0

], . . . ,Γn =

[0 0 0 . . . 1 0 00 0 0 . . . 0 2 00 0 0 . . . 0 0 3

]

is also g-Riesz sequence for C3n with respect to C3. Then the dual g-Riesz sequence of Γjj∈I is:

Γ1 =

[1 0 0 . . . 0 0 00 1

2 0 . . . 0 0 00 0 1

3 . . . 0 0 0

], . . . , Γn =

[0 0 0 . . . 1 0 00 0 0 . . . 0 1

2 00 0 0 . . . 0 0 1

3

]

The sequence Θii∈I for the sequence Λ with respect to (Ξ, Γ) is defined as follow:

Θ1 =

[1 0 0 . . . 0 0 00 −1

2 0 . . . 0 0 00 0 −1

3 . . . 0 0 0

], . . . ,Θn =

[0 0 0 . . . 1 0 00 0 0 . . . 0 −1

2 00 0 0 . . . 0 0 −1

3

]

The following theorem shows relating between the sequences.

theorem 2.3. Let Λ = Λii∈I be a g-Bessel sequence for H with respect to Vii∈I and Γ = Γjj∈I bea g-Riesz sequence for H with respect to Wjj∈I . Let Ξ = Ξii∈I be a g-orthonormal base for H withrespect to Wii∈I . Then for all i, j ∈ I

ΛiΞ∗j = ΘiΓ

∗j ∀i, j ∈ I.

theorem 2.4. Suppose that Γii∈I is a g-Riesz base for a closed subspace W of H with g-Riesz boundsA1, B1, and let Ξii∈I be a g-orthonormal base for H with respect to Wii∈I . The sequence Θii∈I isdefined by (15) for a given sequence Λii∈I . Then we have the following :

(i) If Θii∈I is a g-Bessel forW with respect to Vii∈I with g-Bessel bound B, then Λii∈I is a g-Besselsequence for H with g-Bessel bound BB1.

(ii) If Θii∈I is a g-frame for W with respect to Vii∈I with g-frame bounds A and B, then Λii∈I is ag-frame for H with g-frame bounds AA1, BB1.

(iii) If Θii∈I is a g-Riesz basis for W with respect to Vii∈I with g-Riesz bounds A, B, then Λii∈I isa g-Riesz basis for H with g-Riesz bounds AA1, BB1.

theorem 2.5. Suppose that Γii∈I is a g-Riesz base for a closed subspace W of H with g-Riesz boundsA1, B1, and let Ξii∈I be a g-orthonormal base for H with respect to Wii∈I . The sequence Θii∈I isdefined by (15) for a given sequence Λii∈I . Then we have the following :

(i) If Λii∈I is a g-Bessel for H with respect to Vii∈I with g-Bessel bound B, then Θii∈I is a g-Besselsequence for W with g-Bessel bound B

A1.

(ii) If Λii∈I is a g-frame for H with respect to Vii∈I with g-frame bounds A and B, then Θii∈I is ag-frame for W with g-frame bounds A

B1, BA1.

(iii) Assume that Λ = Λii∈I is a g-Bessel sequence for H with respect to Vii∈I . Then for each f ∈ H,we have

∥∥∥∑

i∈IΘ∗i gi

∥∥∥2≤ B1

A21

∥∥∥∑

i∈IΛ∗i gi

∥∥∥2

and∥∥∥∑

i∈IΛ∗i gi

∥∥∥2≤ 1

A1

∥∥∥∑

i∈IΘ∗i gi

∥∥∥2

(iv) If Λii∈I is a g-Riesz basis for H with respect to Vii∈I with g-Riesz bounds A,B, then Θii∈I isa g-Riesz basis for H with respect to Vii∈I with g-Riesz bounds AA1 and BB1

A21.

65


Our first goal is to characterize the g-R-dual of a given g-Bessel sequence Λii∈I for H with respectto Vii∈I . The idea of our approach is to assume that Ξii∈I is a given g-orthonormal basis and themcharacterize the sequences that can be used in place of Ψii∈I in (14) ; the hope is that at least one ofthem will be a g-orthonormal basis for H.

theorem 2.6. Let Ξii∈I be a g-orthonormal base for H with respect to Wii∈I and Let Λii∈I bea g-Bessel sequence for H with respect to Vii∈I . Suppose that Γjj∈I is a g-Riesz sequence for W =spanΓ∗j (Wj)j∈I . Then the following statements hold:

(i) There exists a sequence Ψii∈I in H such that

Λi =∑

j∈IΨiΓ

∗jΞj ∀i ∈ I. (16)

(ii) The sequences Ψii∈I satisfying (16) are characterized as

Ψi = Ωi + Θi (17)

where Θi is given by (15) and Ωi ∈W⊥ satisfies Ωif = 0 for any f ∈ W and any i ∈ I.

(iii) If Γii∈I is a g-Riesz basis for H, then Ψi = Θi ,i ∈ I, is the unique solution to (17).

theorem 2.7. Let Γjj∈I be a g-Riesz sequence for W = spanΓ∗j (Wj)j∈I and let Ξii∈I be a g-orthonormal basis for H with respect to Wii∈I . Suppose that Λii∈I is a g-frame for H. Then thefollowing statements are equivalent:

(i) There exists a g-orthonormal basis Ψii∈I for H with respect to Vii∈I satisfying (2).

(ii) The sequence Θii∈I in (4) is a Parseval g-frame for W .

theorem 2.8. Suppose Λ = Λii∈I is a g-frame for H with respect to Vii∈I and Γ = Γjj∈I is a g-Rieszbasis for W = spanΓ∗j (Wj)j∈I . Then the following are equivalent :

(i) There exists an g-orthonormal basis Ξii∈I such that Θii∈I is a Parseval g-frame.

(ii) There exists a unitary operator ∆ such that SΓ = ∆SΛ∆∗, where SΛ is the g-frame operator ofΛ = Λii∈I .

corollary 2.9. Suppose Λ = Λii∈I is a tight g-frame for H with respect to Vii∈I and Γ = Γjj∈I isa tight g-Riesz sequence in H. Λii∈I is a g-R-dual of Γjj∈I if and only if the following two conditionshold:

(i) there exists an unitary operator ∆ such that SΓ = ∆SΛ∆∗.

(ii) dim(Span(Γj)∗(Wj)j∈I

)⊥= dim kerT ∗Λ.

Acknowledgment

The authors would like to thank the anonymous reviewers for carefully reading of the manuscript and givinguseful comments, which will help to improve the paper.

66


References

[1] A. Abdollahi, E. Rahimi, Generalized frames on super Hilbert spaces. Bull. Malys. Math. Sci. Soc. 35(3) (2012), 807-818.

[2] M. S. Asgari, Operator-valued bases on Hilbert spaces. J. Linear and Topological Algebra., Vol. 2, No.4, (2013) 201-218.

[3] P. G. Casazza, G. Kutyniok, M. C. Lammers, Duality principles in frame theory. J. Fourier Anal. Appl.,10 (4) (2004), 383408.

[4] O. Christensen, An introduction to frames and Riesz bases. Springer, Switzerland, 2016.

[5] O. Christensen, H. O. Kim, R. Y. Kim, On the Duality Principle by Casazza, Kutyniok, and Lammers.J. Fourier Anal. Appl., 17 (4) (2011), 640-655.

[6] F. Enayati, M. S. Asgari, Duality properties for generalized frames. Banach J. Math. Anal., 11 (4) (2017),880-898.

[7] D. T. Stoeva, O. Christensen, On R-Duals and the Duality Principle in Gabor Analysis. J. Fourier Anal.Appl., 21 (2) (2015), 1531-5851.

[8] D. T. Stoeva, O. Christensen, On Various R-duals and the Duality Principle. Integr. Equ. Oper. Theory84 (2016), 577-590.

[9] W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl. 322 (2006), 437-452.

[10] W. Sun, Stability of g-frames, J. Math. Anal. Appl., 326 (2007), 858-868.

[11] Y. C. Zhu, Characterizations of g-frames and g-Riesz bases in Hilbert spaces. Acta Math. Sinica, 24(10) (2008), 1727-1736.

67

A note on Bochner-Schenberg-Eberlin algebras

E. Feizi* and Z. Sadat Hoseini

(Bu-Ali Sina University, Hamedan, Iran)

Abstract: In this note, we discus on BSE-algebras and present vector-valued weighted Banach space ofholomorphic functions Hυ(D,A) as an example of these algebras.

keywords. Weight, Holomorphic function, Weighted Banach algebras of holomorphic functions, BSE-algebra.


1 Introduction

Let A be a commutative unital semisimple Banach algebra and ΦA be maximal ideal space of A endowedwith the weakest topology with respect to which all the functions

ΦA → C, ϕ 7→ ϕ(a), (a ∈ A)

are continuous. This topology on ΦA is called the Gelfand topology. The norm-decreasing homomorphismG : A → C(ΦA); a 7→ a is called the Gelfand transform of a and the homomorphism G is the Gelfandrepresentation of A. The range of this map is also denoted by A. Cb(ΦA) stands for the Banach algebra ofbounded continuous functions on ΦA with supremum norm.A bounded linear operator T is called multiplier if x(Ty) = (Tx)y for all x, y ∈ A. The set of all themultipliers of A is denoted by M(A). Note that M(A) is a commutative unital Banach subalgebra of B(A)which contains A as a subalgebra (a 7→Ma,Ma(b) = ab).

Now we interest to study a special subspace of Cb(ΦA) which contains A as a subspace. To introducethis space, we need the following definition.

definition 1.1. Let σ be a bounded continuous complex-valued function on ΦA which satisfies in thefollowing: there exists a constant β > 0 such that for any finite number ϕ1, ϕ2, . . . , ϕn in ΦA and of complexnumbers c1, c2, . . . , cn, the next inequality holds,

|∑ni=1 ciσ(ϕi)| ≤ β‖

∑ni=1 ciϕi‖A∗ .

This inequality is called the Bochner-Schoenberg-Eberlein (BSE)-inequality and σ is called BSE-function.The set of all BSE-functions is denoted by CBSE(ΦA) which is a semisimple commutative Banach algebrawith respect to the BSE-norm as infimum of all above β.

A commutative Banach algebra A is called without order if for all a ∈ A, aA = 0 implies that a = 0.

definition 1.2. A without order commutative Banach algebra A is called a BSE-algebra if

68

A note on Bochner-Schenberg-Eberlin algebras

M(A) = CBSE(ΦA)

where M(A) = T | T ∈M(A).For more discussion on BSE-algebras, see [8, 10, 11] and references therein. The following theorem is

useful to recognize the elements of BSE-functions for Banach algebra A.

theorem 1.3. ([10], Theorem 4) CBSE(ΦA) equals the set of all bounded continuous functions σ on ΦA forwhich there exists a bounded net xλ in A such that limλ xλ(ϕ) = σ(ϕ) for all ϕ ∈ ΦA

There are considerable examples of these algebras. Kaniuth and Ulger in ([8], Theorem 2.6.) provedevery uniform algebra is a BSE-algebra. Takahasi and Hatori in [10] also gave considerable examples ofthese algebras such as commutative C∗-algebras, disc algebra A(D) and Hardy algebra Hp(D). In the fol-lowing, we decide to introduce another example of BSE-algebras, vector-valued weighted Banach algebra ofholomorphic functions.

Let D be the open unit ball of the complex plain C. A weight υ : D → (0,∞) is a continuous strictlypositive function. The function υ is called radial when υ(z) = υ(λz) for each λ ∈ C with | λ |= 1. Theweighted spaces of vector-valued holomorphic functions Hυ(D,A) associated with υ and Banach algebra Ais defined by

Hυ(D,A) := f ∈ H(D,A) :‖ f ‖υ= supz∈D ‖f(z)‖Aυ(z) <∞where H(D,A) stand for the space of all holomorphic functions on D. For more information, see [2], [7] and[9].

Let U be an open subset of complex Banach space X. A set A ⊆ U is said to be a U -bounded set if itis bounded and d(A,X \ U) > 0. Following the definition in [5], a family V of weights satisfies condition Iif for every U -bounded set A there exists some υ ∈ V such that infz∈A υ(z) > 0.

Daniel Carando and Pablo Sevilla-Peris [4] proposed necessary and sufficient conditions that HV (U) bean algebra for a family of radial bounded weights V satisfying Condition I.

proposition 1.4. ([4]) Let U be an open and balanced subset of Banach space X and V be a family ofradial bounded weights satisfying Condition I. Then HV (U) is an algebra if and only if for every υ thereexist ω ∈ V and c > 0 so that

υ(x) ≤ cω2(x) (x ∈ U).

We clime that Hυ(D,A) ia an algebra and so a Banach algebra when there exists λ > 0 such that υ ≥ λ.It is only need to give V = υ and put c := 1

infz∈D υ(z) in last proposition. We call this weight an algebraweight.

2 The BSE-algebra property of Hυ(D,A)

We assume that υ be an algebra weight. We show that Hυ(D,A) is a BSE-algebra with some conditions.

As we saw, every uniform algebra is a BSE-algebra, whereas every Banach function algebra (BFA) isnot. Takahasi and Hatori demonstrated when a BFA is a BSE-algebra.

theorem 2.1. ([3], Nearly Takahasi and Hatori) A BFA A on a compact space K is a BSE algebra if andonly if the closed unit ball BA is closed in the topology of pointwise convergence on K.

Gupta and Baweja in [6] introduced a new topology on Hυ(U,A), τM , for an open subset U of a Banachspace E with values in a Banach algebra A, such that τco ≤ τM ≤ τ‖.‖υ for compact-open topology τcoand they proved that τM = τco on ‖.‖υ-bounded set B. Let P(mE,F ) be the space of all continuous m-homogeneous polynomials for two Banach spaces E and F . A continuous polynomial P is a mapping from Einto F which can be represented as a sum P = P0 +P1 + · · ·+Pk with Pm ∈ P (mE,F ) for m = 0, 1, · · · , k.The vector space of all continuous polynomials from E into F is denoted by P(E,F ). However, theypresented the following theorem where P(E) equals P(E,C).

69

E. Feizi, Z. Sadat Hoseini

proposition 2.2. ([6], Proposition 4.6.) Let E and F be Banach spaces. For a radial weight υ on a balancedopen subset U of E with P(E) ⊆ Hυ(U), the space P(E,F ) is τM -dense in Hυ(U,F ).

Now we prove cardinal theorem.

theorem 2.3. Let D be the open unit ball in the complex plain, υ be a radial weight and A be a commutativeunital semisimple Banach algebra with P(C) ⊆ Hυ(D). Then Hυ(D,A) is a BSE-algebra.

Proof. Hυ(D,A) is a BFA where A is a commutative unital semisimple Banach algebra. By proposition2.2, Hυ(D,A) and so BHυ(D,A) is τM -closed. However, τco |B= τM |B for closed unit ball B(:= BHυ(D,A)) ofBanach algebra Hυ(D,A). Hence BHυ(D,A) is τco-closed. On the other hand, by Vitali theorem ([1], Theorem2.1) in vector-valued case, τpw = τco for pointwise convergence topology τpw on B. The proof is complete byinvoking Theorem 2.1

It is remarkable that for weight function υ with υ ≥ λ (λ > 0), Hυ(D,A) is an algebra. Furthermore,P(C) ⊆ Hυ(D) implies that υ is bounded from above. These two conditions result Hυ(D,A) = H∞(D,A),the algebra of all bounded holomorphic mappings, and then H∞(D,A) is a BSE-algebra. Although it iseasy to see that scalar-valued Banach algebra H∞(D) is also a BSE-algebra, since it is a uniform algebra.

References

[1] W. Arendt and N. Nikolski, Vector-valued holomorphic functions revisited, Math. Z. 234 (2000), pp.777-805.

[2] J. Bonet, P. Domansky, M. Lindstrom and J. Taskinen, Composition operators between weighted spacesof analytic functions, J. Austral. Math. Soc. 64 (1998), pp. 101-118.

[3] H. G. Dales and Lancaster, Approximate identities and BSE norms for Banach function algebras, FieldsInstitute, Toronto. (2014), pp. 1-38.

[4] Daniel Carando and Pablo Sevilla-Peris, Spectra of weighted algebras of holomorphic functions, MathZ. 263 (2009), pp. 887-902.

[5] D. Garca, M. Maestre, P. Rueda, Weighted spaces of holomorphic functions on Banach spaces, StudiaMath. 138, n. 1, (2000), pp. 1-24.

[6] Manjul Gupta and Deepika Baweja, Weighted Spaces of Holomorphic Functions on Banach Spaces andthe Approximation Property, Extracta Mathematicae Vol. 31. (2016), pp. 123-144.

[7] Enrique Jorda, Weighted Vector-Valued Holomorphic Functions on Banach Spaces, Hindawi PublishingCorporation Abstract and Applied Analysis. (2013), pp. 110.

[8] E. Kaniuth and Ali Ulger, The Bochner-Schoenberg-Eberlein property for commutative Banach algebras,especially Fourier and Fourier-Stieltjes algebras, Trans. Amer. Math. Soc. 362 (2010), pp. 43314356.

[9] Jasbir Singh Manhas, Weighted composition operators on weighted spaces of vector-valued analyticfunctions, J. Korean Math. Soc. 45 (2008), No. 5, pp. 1203-1220.

[10] Sin-Ei Takahasi and Osamu Hatori, Commutative Banach algebras which satisfy a Bochner-Schenberg-Eberlin-type theorem, Proceedings of AMS. 110 (1999), pp. 149-158.

[11] S.-E. Takahasi, Y. Takahashi, O. Hatori and K. Tanahashi, Commutative Banach algebras and BSE-norm, Math. Japonica. 46 (1997), pp. 273-277.

70

Centralizing derivations on Banach algebras related to locally compactgroups

M. Ghasemi* and M. J. Mehdipour

(Shiraz University of Technology, Shiraz, Iran)

Abstract: Let G be a locally compact group and A be the Banach algebra L∞(G)∗ or LUC(G)∗. Inthis paper, we investigate (skew) centralizing derivations on A and show that they map A into C0(G)⊥A .We also prove that if d is a derivation on L∞(G)∗ with d(F ·H)±F ·H ∈ Z(L∞(G)∗) for all F,H ∈ L∞(G)∗,then G is isomorphic to Zn for some n ∈ Z.

keywords. locally compact group, derivation, centralizing map.

subject. 43A15, 16W25.

1 Introduction

Let G denote a locally compact group with a fixed left Haar measure λ. The group algebra L1(G) is definedas in [2] equipped with the convolution product “∗” and the norm ‖.‖1 Also, let L∞(G) denote the usualLebesgue space as defined in [2] equipped with the essential supremum norm ‖.‖∞. Then L∞(G) is the dualof L1(G) for the pairing

〈f, φ〉 =

∫

Gf(x)φ(x)dλ(x)

for all φ ∈ L1(G) and f ∈ L∞(G). It is well-known from [1] that the dual of L∞(G), represented by L∞(G)∗,is a Banach algebra with the first Arens product “·” defined by

〈F ·H, f〉 = 〈F,Hf〉,

where

〈Hf, φ〉 = 〈H, fφ〉 and 〈fφ, ψ〉 = 〈f, φ ∗ ψ〉for all F,H ∈ L∞(G)∗, f ∈ L∞(G) and φ, ψ ∈ L1(G).

We denote by C(G) the space of all bounded continuous complex-valued functions on G with the sup-norm and let C0(G) be the subalgebra of C(G) consisting of all functions that vanish at infinity. Wedenote by LUC(G) the Banach space of all f ∈ C(G) such that the mapping x 7→ lxf from G into C(G) iscontinuous, where

lxf(y) = f(xy)

for all x, y ∈ G. The Banach space LUC(G)∗ is also a Banach algebra with the Arens product defined by

〈µ ν, f〉 = 〈µ, νf〉,

71

M. Ghasemi, M. J. Mehdipour

in which〈νf, x〉 = 〈ν, lxf〉

for all µ, ν ∈ LUC(G)∗, f ∈ LUC(G) and x ∈ G.Let R be an associative ring with center Z(R). Let us recall that R is prime if aRb = (0) implies that

either a = 0 or b = 0. For every a, b ∈ R, we define

[a, b] = ab− ba and a b = ab+ ba.

For a positive integer k, an additive map d : R→ R is called k−(skew)centralizing if for every a ∈ R

[d(a), ak] ∈ Z(R), (d(a) ak ∈ Z(R)).

In a special case when[d(a), ak] = 0, (d(a) ak = 0),

d is called k−(skew) commuting. Also, d is called a derivation if for every a, b ∈ R,

d(ab) = d(a)b+ ad(b).

The singer-Wermer theorem, which is a classical theorem of Banach algebra theory, states that everycontinuous derivation on a commutative Banach algebra maps into its Jacobson radical [5], and Thomas[6] proved that the Singer-Wermer theorem remains true without assuming the continuity of the derivation.(This generalization is called the Singer-Wermer conjecture). On the other hand, Posner [2] obtained twofundamental results in 1957: (i) the first result (the so-called posner’s first theorem) asserts that if d and gare derivations on a 2-torsion free prime ring such that the product dg is a derivation, then either d = 0 org = 0. (ii) The second result (the so-called Posner’s second is a centralizing derivation on a noncommutativeprime ring, then d = 0. Mathieu and Runde [2] generalized the Singer-Wermer conjecture by proving thatevery centralizing derivation on a Banach algebra maps into its Jacobson radical.

In this paper, we investigate centralizing derivations on Banach algebras L∞(G)∗ and LUC(G)∗ and giveanalogues of Posner’s second theorem.

2 Main results

Let A be the Banach algebra L∞(G)∗ or LUC(G)∗. Let also

C0(G)⊥A := m ∈ A : m|C0(G) = 0.

theorem 2.1. Let G be a locally compact group and d be a (skew) centralizing derivation on A. Then therange of d can be embedded into C0(G)⊥A .

corollary 2.2. Let G be a locally compact group and let d be a 2 -centralizing derivation on A. Then therange of d can be embedded into C0(G)⊥A .

theorem 2.3. Let G be a compact group and d be a (skew) centralizing derivation on A. Then d = 0 on A.

Let n be a fixed integer. The additive group of integers modulo n is denoted by Zn.

corollary 2.4. Let G be a compact group and let d be a derivation on L∞(G)∗ with

d(F ·H)± F ·H ∈ Z(L∞(G)∗)

for all F,H ∈ L∞(G)∗. Then G is isomorphic to Zn for some n ∈ Z.

We finish the paper with the following result.

proposition 2.5. Let G be a locally compact group and d be a derivation on LUC(G)∗ with

d(µ ν)± µ ν ∈ Z(LUC(G)∗)

for all µ, ν ∈ LUC(G)∗. Then d is commuting.

72

Centralizing derivations on Banach algebras related to locally compact groups

References

[1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc., 2 (1951), 939-948.

[2] E. Hewitt, K. Ross, Abstract Harmonic Analysis I, Springer-Verlag, New-York, 1970.

[3] M. Mathieu, V. Runde, Derivations mapping into the radical II, Bull. London Math. Soc., 24 (1992),485-487.

[4] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100.

[5] I. M. Singer, J. Wermer, Derivations on commutative normed algebras, Ann. Math., 129 (1955), 260-264.

[6] M. P. Thomas, The image of derivation is containd in the radical, Ann. Math., 128 (1988), 435-460.

73

Properties of g-frame representations with bounded operators

F. Ghobadzadeh* and Y. Khedmati

(Mohaghegh Ardabili University, Ardabil, Iran)

Abstract: Christensen et al., study frames for H with index set Z, that have representations inthe form T if0i∈Z. The purpose of this paper is to study some properties of T ∈ GL(H) for g-framesΛ = Λi ∈ B(H,K) : i ∈ Z having the form Λi = Λ0T

i.

keywords. representation, perturbation, g-frame

subject. 41A58, 42C15, 47A05

1 Introduction

In 1952, the concept of frames for Hilbert spaces was introduced by Duffin and Schaeffer. Throughoutthis paper, I is a countable index set, H and K are seperable Hilbert spaces, Ki : i ∈ I is a familyof seperable Hilbert spaces, IdH denotes the identity operator on H, B(H) and GL(H) denote the setof all bounded linear operators and the set of all invertible bounded linear operators on H, respectively,l2(H, I) =

gii∈I : gi ∈ H,

∑i∈I ‖gi‖2 <∞

as well. Also, we will apply B(H,K) for the set of all bounded

linear operators from H to K.Aldroubi et al., introduced the concept of dynamical sampling deals with frame properties of sequences

of the form T if1i∈N, for f1 ∈ H and T : H → H belongs to certain classes of the bounded operators[1]. Christensen and Hassannasab analyze frames F = fii∈Z having the form F = T if0i∈Z, where T isa bijective linear operator (not necessarily bounded) on Spanfii∈Z [2]. They also stated a perturbationcondition that preserves the existence of a representation F = T if0i∈Z.

proposition 1.1. [2] Assume that F = fii∈Z = T if0i∈Z is a frame for H and let gii∈Z be a sequencein H. Assume that there exist constants λ1, λ2 ∈ [0, 1) such that

∥∥∑ ci(fi − gi)∥∥ ≤ λ1

∥∥∑ cifi∥∥+ λ2

∥∥∑ cigi∥∥

for all finite sequences cii∈Z. Then G = gii∈Z is a frame for H; furthermore G = V ig0i∈Z for a linearoperator V : SpanT if0i∈Z → SpanT if0i∈Z. If T is bounded, then V is also bounded.

In 2006, generalized frames (or simply g-frames) and g-Riesz bases were introduced by Sun [5].

definition 1.2. We say that Λ = Λi ∈ B(H,Ki) : i ∈ I is a generalized frame, or simply g-frame, for Hwith respect to Ki : i ∈ I if there are two constants 0 < AΛ ≤ BΛ <∞ such that

AΛ‖f‖2 ≤∑

i∈I‖Λif‖2 ≤ BΛ‖f‖2, f ∈ H. (18)

We call AΛ, BΛ the lower and upper g-frame bounds, respectively. Λ is called a tight g-frame if AΛ = BΛ.

74

Properties of g-frame representations with bounded operators

For a g-frame Λ, there exists a unique positive and invertible operator SΛ : H → H such that

SΛf =∑

i∈IΛ∗iΛif, f ∈ H,

and AΛIdH ≤ SΛ ≤ BΛIdH. Consider the space

(∑

i∈I⊕Ki

)l2

=fii∈I : fi ∈ Ki, i ∈ I and

∑

i∈I‖fi‖2 <∞

.

For a g-frame Λ, the synthesis operator TΛ :(∑

i∈I ⊕Ki)l2→ H is defined by

TΛ

(gii∈I

)=∑

i∈IΛ∗i gi, gi ∈ Ki.

The adjoint of TΛ, T ∗Λ : H →(∑

i∈I ⊕Ki)l2

is called the analysis operator of Λ and is as follow

T ∗Λf = Λifi∈I , f ∈ H.

It is obvious that SΛ = TΛT∗Λ.

In [3], we generalize some recent results of Christensen et al., [2] to investigate g-frames Λ = Λi ∈B(H,K) : i ∈ I (I = N or Z) having the form Λi+1 = Λ1T

i with T ∈ B(H). In this paper, we studystability of g-frame representations under some perturbations.

2 Stability of g-frame representations

In [3], we study the concept of representation for g-frames with bounded operators and some results forit. Sun has proved that g-frames are stable under small perturbations and has studied the stability of dualg-frames [6]. You can find more perturbation results for g-frames in [4]. In [2], It has stated a perturbationcondition that preserves the existence of a representation for a frame. In this section, we study stability ofg-frame representations under some perturbations.

definition 2.1. We say that a g-frame Λ = Λi ∈ B(H,K) : i ∈ N has a representation, if there is aT ∈ B(H) such that Λi = Λ1T

i−1, i ∈ N, and we say that a g-frame Λ = Λi ∈ B(H,K) : i ∈ Z has arepresentation, if there is a T ∈ GL(H) such that Λi = Λ0T

i, i ∈ Z. In the affrimative case, we say that Λis represented by T .

We will need the right-shift operator on l2(H,N) and l2(H,Z), defined by T (cii∈N) = (0, c1, c2, ...) andT (cii∈Z) = ci−1i∈Z. By generalizing a result of the [2], the following theorem give sufficient conditionsfor g-frame Λ = Λi ∈ B(H,K) : i ∈ N to have a representation.

theorem 2.2. [3] Let Λ = Λi ∈ B(H,K) : i ∈ N be a g-frame that for any finite set In ⊂ N, andgii∈In ⊂ K,

∑i∈In Λ∗i gi = 0 implies gi = 0 for any i ∈ In. Suppose that kerTΛ is invariant under the

right-shift operator T . Then, Λ is represented by T ∈ B(H), where ‖T‖ ≤√BΛA

−1Λ .

Every g-orthonormal basis and g-Riesz basis have representations [3].

Note that, the Theorem 2.2 can be satisfied for g-frames with index set Z, as well. In the reminder, wewant to give our main results about the stability of g-frame representations under some perturbations.

theorem 2.3. Let Λ = Λi ∈ B(H,K) : i ∈ I (I = N or Z) and Θ = Θi ∈ B(H,K) : i ∈ I be twog-frames such that kerTΛ = kerTΘ. Then, Λ has a representation if and only if, Θ has a representation.

75

F. Ghobadzadeh, Y. Khedmati

Proof. The operators TΛ and TΘ are onto [4, Proposition 2.6], and so for any f ∈ H, there is aii∈I ∈ l2(K, I)such that TΛaii∈I = f . We define the operator U ∈ B(H) by Uf = TΘaii∈I . By kerTΛ = kerTΘ, U iswell-defined and injective. On the other haned, TΘ is onto, and so U is onto. Therefore, U ∈ GL(H). Forany gii∈I ∈ l2(K, I) and g ∈ H ,we have

⟨gii∈I ,

(Θi − ΛiU

∗)gi∈I

⟩=∑

i∈I

⟨gi, (Θi − ΛiU

∗)g⟩

=∑

i∈I

⟨Θ∗i gi, g

⟩−∑

i∈I

⟨Λ∗i gi, U

∗g⟩

=⟨TΘgii∈I , g

⟩−⟨UTΛgii∈I , g

⟩

=⟨UTΛgii∈I , g

⟩−⟨UTΛgii∈I , g

⟩

= 0,

therefore Θi = ΛiU∗. If Λ is represented by T , then Θ is represented by (U∗)−1TU∗, in fact we have

Θi(U∗)−1TU∗ = ΛiU

∗(U∗)−1TU∗ = ΛiTU∗ = Λi+1U

∗ = Θi+1.

The other implication is the same.

corollary 2.4. Suppose that a g-frame Λ = Λi ∈ B(H,K) : i ∈ I, (I = N or Z) has a representation andΘ = Θi ∈ B(H,K) : i ∈ I is a family of operators such that for any finite index set In ⊂ I

∥∥∑

i∈In(Λi −Θi)

∗gi∥∥ ≤ λ

∥∥∑

i∈InΛ∗i gi

∥∥+ µ∥∥∑

i∈InΘ∗i gi

∥∥, gi ∈ K, (19)

where 0 ≤ maxλ, µ < 1. Then, the family Θ is a g-frame that has a representation.

Proof. The family Θ is a g-frame [4, Theoerem 3.5], and so by the inequlity (19), we get kerTΛ = kerTΘ.By the Theorem 2.3, the proof is completed.

corollary 2.5. Suppose that a g-frame Λ = Λi ∈ B(H,K) : i ∈ I, (I = N or Z) has a representation andΘ = Θi ∈ B(H,K) : i ∈ I is a g-frame such that for a constant C > 0,

∥∥∑

i∈I(Λi −Θi)

∗gi∥∥2 ≤ C.min

∥∥∑

i∈IΛ∗i gi

∥∥2,∥∥∑

i∈IΘ∗i gi

∥∥2, gii∈I ∈ l2(K, I). (20)

Then, Θ has a representation.

Proof. By the inequlity (20), It is obvious that kerTΛ = kerTΘ. Then, by the Theorem 2.3, Θ has arepresentation.

References

[1] A. Aldroubi, C. Cabrelli, U. Molter, and S. Tang, Dynamical sampling, Applied and ComputationalHarmonic Analysis, 42(3) (2017), pp. 378–401.

[2] O. Christensen, and M. Hasannasab, Operator representations of frames: boundedness, duality, andstability, Integral Equations and Operator Theory, 88(4) (2017), pp. 483-499.

[3] Y. Khedmati and F. Ghobadzadeh, G-frame representation with bounded operators, arXiv:1812.00386v1.

[4] A. Najati, M. H. Faroughi, and A. Rahimi, G-frames and stability of g-frames in Hilbert spaces, Methodsof Functional Analysis and Topology, 14(03) (2008), pp. 271-286.

[5] W. Sun, G-frames and g-Riesz bases, Journal of Mathematical Analysis and Applications, 322(1) (2006),pp. 437–452.

[6] W. Sun, Stability of g-frames, Journal of mathematical analysis and applications, 326(2) (2007), pp.858–868.

76

On the uniform distribution of sequences involving the imaginary partsof the non-trivial zeros of the Riemann zeta function

M. Hassani* (University of Zanjan, Zanjan, Iran)

Abstract: In this paper we provide a geometric study of the uniform distribution of several sequencesinvolving the sequence γn, the imaginary parts of the non-trivial zeros of the Riemann zeta function. Wegenerate several graphs of the Weyl sums involving the sequence (αγn)n≥1, for certain values of α, as wellas, several sequences concerning the values of γn. We use a theorem of Landau to explain some appearedgeometric patterns.

keywords. The Riemann zeta function, Distribution modulo one, Weyl sums

subject. 11M26, 11J71, 11K06, 11L15

1 Introduction: Uniform distribution and Weyl’s Criterion

Among several interfaces between analytic number theory and harmonic analysis (see [12] for a number ofthem), we read about uniform distribution of sequences, a tool to detect more details of the distribution ofthe values of number theoretic sequences.

An arbitrary real sequence (an)n≥1 is uniformly distributed modulo 1, if for all real numbers a, b with0 ≤ a < b ≤ 1 we have #n ≤ N : an ∈ [a, b] ∼ (b − a)N as N → ∞. Here, x = x − bxc denotes thefractional part of the real number x. An efficient criterion to determine uniform distribution modulo 1 ofa given sequence, due to Weyl [18], asserts that the sequence (an)n≥1 is uniformly distributed modulo 1 ifand only if

∑n≤N e(han) = o(N) as N →∞, for every positive integer h. Here, and in what follows, we let

e(x) = e2πix.

Uniform distribution modulo 1 of various number theoretic functions is known in literature. Among them,Rademacher [15] observed that under the assumption of the Riemann Hypothesis the sequence (αγn)n≥1 isuniformly distributed modulo 1, where α 6= 0 is a fixed real number and γn runs over the imaginary partsof the zeros of ζ(s). Later Hlawka [8] proved this assertion unconditionally.

2 Geometrical approach

Dekking and Mendes France [2] introduced an idea to make visible the Weyl sums∑

n≤N e(han) for agiven real sequence (an)n≥1 and given positive integers h and N , by drawing a plane curve generated bysuccessively connected lines segment, which joins the point Vn to Vn+1, where Vn = (S1(n), S2(n)) withS1(n) =

∑k≤n cos(2πhak) and S2(n) =

∑k≤n sin(2πhak), for 1 ≤ n < N . Some classical examples are

pictured in [2] and [3].

In this paper, which is essentially a reproduction of [7], we generate several graphs of the Weyl sumsinvolving the sequence (αγn)n≥1, for certain values of α, as well as, several sequences concerning the values

77

M. Hassani

0 5 10 15 20

K10

K5

0

5

10

15

K20 0 20 40 60 80

K40

K30

K20

K10

0

10

20

30

40

50

K20 K10 0 10 20 30 40K60

K40

K20

0

20

40

Figure 1: Graphs of the Weyl sums∑

n≤5000 e(αγn) for α = 1, π, e, respectively from left to right

K60 K50 K40 K30 K20 K10

K1.00.01.0

K180 K160 K140 K120 K100 K80 K60 K40 K20

K1.0

1.0

K400 K300 K200 K100

K1.0

1.0


n≤N e(αγn) with α = (log 2)/2π, for N = 500, 2000, 5000 respectivelyfrom top to down

of γn. To generate figures appeared on this paper, we used Maple software to do several computationsrunning over the numbers (γn)1≤n≤20000, all of which are based on the tables of zeros of the Riemann zetafunction due to Odlyzko [13].

We note that in the graphs of the Weyl sums, the length of each line segment is 1. Thus, if 1 ≤n < N , then the frame (rectangular border) that includes the graph of the Weyl sum has the size notexceeding N ×N . We denote the height and the width of the frame containing the graph of the Weyl sum∑

n≤N e(han), respectively by Hh(N) and Wh(N). Since∑

n≤N e(han) = S1(N) + iS2(N), then the Weylcriterion geometrically asserts that the real sequence (an)n≥1 is uniformly distributed modulo 1 if and only ifHh(N) = o(N) and Wh(N) = o(N) as N →∞, for any positive integer h. Since it is not possible to considerall positive integer values of h, hence we will take h = 1 in all graphs, and simply we let H1(N) = H(N)and W1(N) = W (N). In this paper we study two different families of the graphs of the Weyl sums involvingthe sequence (αγn)n≥1 according to the ratio H(N)/W (N) of the frame that includes the graph of the Weylsum, as in Figure 1, and Figure 2.

We focus on the interesting case α = (logm)/2π, where m is a positive integer. In Section 3 we willexplain why this case is interesting in our investigation. Thanks to the works of Rademacher [15] andHlawka [8], all of the sequences we pictured their Weyl sums in this section and in Section 3, are uniformlydistributed modulo 1. More precisely, Hlawka [8] proved that the discrepancy of the set αγn : 0 < γn ≤ Twith α = (logm)/2π, is O(1/ log T ), under the assumption of the Riemann Hypothesis, and that it isO(1/ log log T ) unconditionally. Fujii [6] showed unconditionally that this discrepancy is O(log log T/ log T ).We recall that the discrepancy of the real numbers a1, a2, . . . , aN is defined by

sup0≤a<b≤1

∣∣∣∣1

N#n ≤ N : an ∈ [a, b]

− (b− a)

∣∣∣∣ .

For more information about discrepancy we refer the reader to the second chapter of the book of Kuipersand Niederreiter [10].

We observe that for the case α = (logm)/2π, when m is a prime power (for example as in Figure 2 andFigure 3), the frame including the graph of the Weyl sum has an unusual long size. Of course, this doesn’t

78

n the uniform distribution of sequences involving the imaginary parts of the non-trivial zeros of theRiemann zeta function

K300 K200 K100 0

K1

2

K200 K150 K100 K50 0K3

2

K150 K100 K50 0

K2

5


n≤5000 e(αγn) with α = (log 2k)/2π, for k = 2, 3, 4 respectively fromtop to down

K1 0 1 2 3

K2

K1

0

1

2

K2 K1 0 1 2 3 4 5

K4

K3

K2

K1

0

1

2

3

K2 0 2 4 6 8

K2

0

2

4

6

8


n≤2000 e(αγn) with α = (logm)/2π, for m = 6, 12, 36 respectivelyfrom left to right

happen for other integer values of m, for example in Figure 4, and even for other non-integer values of m,for example in Figure 5. In the next section, we introduce a mathematical justification for this phenomenon,and in the last section, we visualize some more noteworthy patterns.

K2 K1 0 1 2

K3

K2

K1

0

1

2

K4 K3 K2 K1 0 1 2 3

K4

K3

K2

K1

0

1

2

3

K6 K4 K2 0 2 4

K6

K4

K2

0

2

4

K15 K10 K5 0 5 10

K10

K5

0

5

10


n≤2000 e(αγn) with α = (log 21/k)/2π, for k = 2, 3, 4, 10 respectivelyfrom left to right

3 Landau’s formula and justification of the frames’ size

The sizes of the frames including the graphs of the Weyl sums in Section 2 largely explained by a theoremof Landau [11] which asserts, for each fixed real x > 1, that

∑

0<γ≤Txρ = − T

2πΛ(x) +O(log T ),

as T tends to infinity, where ρ = β + iγ runs over the non-trivial zeros of the Riemann zeta function, andΛ(x) is the von Mangoldt function for integral x > 1 and Λ(x) = 0 for non-integral values of x.

79

M. Hassani

Assuming that the Riemann Hypothesis holds, as we did to generate present graphs, one has ρ = 1/2+iγ,and hence ∑

0<γ≤Txiγ = − T

2πx−

12 Λ(x) +O(log T ).

We have miγ = e(αγ) with α = (logm)/2π. As usual, we let N(T ) denote the number of the non-trivialzeros ρ = β + iγ of the Riemann zeta function satisfying 0 < β < 1 and 0 < γ ≤ T . By using the vonMangoldt’s approximation for N(T ) (see [9] and [17]), we obtain the validity of γn ∼ 2πn/ log n as n→∞.Thus, assuming that the Riemann Hypothesis holds, if α = (logm)/2π and N = N(T ), then we obtain

∑

n≤Ne(αγn) = − T

2πm−

12 Λ(m) +O(log T ).

So, for example as in Figure 2, we see that the real part of∑

n≤N e(αγn) tending to −∞, at a rateT ∼= 2πN/ logN , and the imaginary part of it, is bounded by O(log T ). Although in Figure 2 the imaginaryparts are bounded by 1, we expect that eventually they get larger. One will get similar behaviour forα = (logm)/(2π) whenever m is a power of a prime but in other cases, and in particular when m is not aninteger, and consequently Λ(m) = 0, the right-hand-side of the last relation is just O(log T ), and in this caseone would expect a “random walk” of the types in Figure 1, Figure 4, and Figure 5, with the size boundedby O(log T ).

4 Spiral patterns and other patterns

In this section, we produce the graphs of the Weyl sums of several sequences related to the values of γn. Weknow that some of these sequences are uniformly distributed modulo 1, however, we have no mathematicaljustification for the details of the presented patterns. Moreover, we observe that some of them have spiralshapes. To clear the situations of these spiral patterns, we will picture the graphs of the Weyl sums as aspace curve.

4.1 Sequences concerning polynomial values

Figure 6 shows the graphs of the Weyl sums∑

n≤5000 e(γkn) with k = 2, 3, 10. We observe that both of the

ratios H(N)/N and W (N)/N for N = 5000 are small (∼= 0.04). This observation supports the followingconjecture.

Conjecture 1. For any non-constant polynomial P (x) with real coefficients, the sequence (P (γn))n≥1

is uniformly distributed modulo 1.

K120 K100 K80 K60 K40 K20 0

K40

K30

K20

K10

0

10

20

30

K20 0 20 40 60 80

K30

K20

K10

0

10

20

30

40

50

60

K50 K40 K30 K20 K10 0 10

K60

K50

K40

K30

K20

K10

0

10

20


n≤5000 e(γkn) with k = 2, 3, 10 respectively from left to right

4.2 Sequences concerning rational powers

Figures 7 and 8 show the graphs of the Weyl sums∑

n≤5000 e(γrn) for several rational values of r.

80


0 10 20 30 40

0

10

20

30

40

K10 K8 K6 K4 K2 0

K4

K3

K2

K1

0

1

2

3

4

0 10 20 30

K8

K6

K4

K2

0

2

4

6

8

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5

10

15

20

K10 0 10 20 30 40 50

0

10

20

30

40

K20 K15 K10 K5 0 5

K20

K10

0

10


n≤5000 e(γrn) with r = 10/11, 2/3 (first row), r = 3/4, 4/5 (second

row), r = 7/8, 17/20 (third row)

Fujii [5] developed a method to obtain uniform distribution modulo 1 of a family of sequences (f(γn))n≥1

for a wide class of smooth functions f . In particular, his method implies that sequences of the form (γrn)n≥1

for any fixed real r ∈ (0, 1) are uniformly distributed modulo 1. Although in Figure 8 the rectangular frameof the rightmost graph seems large, we expect that eventually it gets smaller to become o(N) as N → ∞,when we consider the related Weyl sum over n ≤ N .

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4.3 Sequences concerning log γn

The above mentioned family of smooth functions f due to Fujii [5], for which the sequences (f(γn))n≥1 areuniformly distributed modulo 1 includes a number of logarithmic functions.

More precisely, he asserts that for any fixed arbitrary real number t > 0 the sequences with generalterms

(log γn)(logk γn), (log γn)1+t,γn

(log γn)t−1,

γn log γnlogk γn

,

all are uniformly distributed modulo 1. Here, k ≥ 1 is an integer, and logk denotes the k-fold iterativelogarithm function. Also, he remarks that the sequence with general term log γn is not uniformly distributed

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modulo 1. Figure 9 shows the graphs of the Weyl sums∑

n≤5000 e(an) with an = log γn, an = γn log γn,an = γn/ log γn and an = n log γn. The sizes of the frames in these graphs led us to the following conjecture.

Conjecture 2. Sequences with general terms an = γn log γn, and an = n log γn are uniformly distributedmodulo 1.

4.4 Sequences concerning the difference γn+1 − γnThe sequence with general term an = (γn+1 − γn)((log γn)/2π)/2π appears in the work of Odlyzko [14],where he studies the distribution of spacing between zeros of the zeta function. He is motivated by the factsthat the non-trivial zeros of ζ(s) become denser and denser the higher up one goes in the critical strip, andthe average vertical spacing between consecutive zeros at height t is 2π/ log(t/2π). Therefore, in order tostudy quantities that are largely invariant with height, he defines, under the assumption of the RiemannHypothesis, the above mentioned sequence an as the normalized spacing between consecutive zeros withimaginary parts γn and γn+1. Moreover, he remarks that the above mentioned sequence an has the meanvalue 1 in the sense that for any positive integers K and L, we have

∑K<n≤K+L an = L+O(log(KL)).

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Figures 10, 11, and 12 show the graphs of the Weyl sums∑

n≤5000 e(an) with an concerning the differenceγn+1−γn in several cases, including the above mentioned sequence studied by Odlyzko in the leftmost graphof Figure 10.

Considering the sizes of the frames in Figures 10, 11, and 12, we formulate the following conjecture.

Conjecture 3. All of the sequences in Figures 10, 11, and 12, except the sequence an = 2π/((log n)(γn+1−γn)), are uniformly distributed modulo 1.

4.5 Tornado patterns in space curves

As we mentioned in the above sections, by a theorem of Fujii, sequences with general terms an = γrnwhere r ∈ (0, 1), and an = γn/ log γn are uniformly distributed modulo 1. Some of the graphs of theircorresponding Weyl sums in Figure 7, and Figure 9, consist of “S” shape, where many of lines segment

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snuggle around two holes and generate S-shape graphs. To detect exactly for which values of N the graphsof the Weyl sums

∑n≤N e(an) generate such S-shapes, we study the space curve form of the graphs of the

Weyl sums∑

n≤N e(han) for a given real sequence (an)n≥1 and given positive integers h and N , by drawinga space curve generated by successively connected lines segment, which joins the point Vn to Vn+1, whereVn = (S1(n), S2(n), n/1000). Here S1(n) =

∑k≤n cos(2πhak) and S2(n) =

∑k≤n sin(2πhak), for 1 ≤ n < N .

In Figure 13, we generate the space curves of the Weyl sums∑

n≤20000 e(an) with an = γ10/11n , an = γ

17/20n ,

and an = γn/ log γn. These curves show more detains of the related S-shape graphs. They contain some“tornado” patterns, and it seems that as N growths in the related Weyl sums

∑n≤N e(an), we expect some

more “tornado” patterns around several holes.

4.6 Further remarks

We end up this paper with some remarks for further investigations and studies of the geometric patternsappeared here.

1. There are also some spiral and tornado patterns in the classical examples, pictured in Figures 14and 15. Studying the mathematical justification for the patterns in these classical examples will help us inunderstanding of the patterns appeared in the case of the Riemann zeta function.

We note that the most simple tornado pattern is the space curve of the Weyl sum∑

n≤N e(an) withan = αn where α is an irrational coefficient. The leftmost graph pictured in Figure 14 shows the space curveof the Weyl sum

∑n≤1000 e(n/e), as an example. The fundamental fact here is that for the linear sequence

an = αn with irrational coefficient α, the generating vertices Vn of the related plane curve lie on a circle withradii 1/(2| sin(πα)|) and the center located at the point (−1/2, cot(πα)/2). Moreover, the graph of the Weylsum of the linear sequence an = αn is dense in an annulus with raduses | cot(πα)|/2 and 1/(2| sin(πα)|).

Based on this fundamental fact, to justify the patterns in Figure 15, Tenenbaum and Mendes France [16]remark that because of the weak growth of log n, the curve behaves locally like the curve associated withthe linear sequence an = cHn where cH is a local constant with cH ≈ logH for n ≈ H. Thus, the graphof the Weyl sum of the sequence an = n log n appears as a succession of annuli, joined by almost straight

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n≤20000 e(an) with an = γ10/11n , an = γ

17/20n , and an =

γn/ log γn, respectively from left to right

lines, corresponding to the values of H such that logH ≈ 0. This will happen for the values of H ≈ em forsome positive integers m satisfying 1 ≤ m ≤ log 10000. In the case of the present example, this will happenat heights H ≈ em with m = 1, 2, . . . , 9, and more visibly for H ≈ 148, 403, 1096, 2980, 8103, as indicated inthe side view in the rightmost graph of Figure 15.

2. One may do similar experiments with other types of L-functions. For example, Akbary and Murty[1] proved that the imaginary parts of the zeros of Dirichlet L-functions, modular L-functions, and MaassL-functions are also uniformly distributed mod 1. So it should be possible to do similar experiments if onehas access to the zeros of these L-functions.

3. As we see, the analysis of the patterns for the sequence (αγn)n≥1 largely depends on the Landau’sformula for the zeros of the Riemann zeta function. Ford and Zaharescu [4] obtained the following uniformversion of this formula

∑

0<γ≤Txρ = −Λ(nx)

2π

eiT log( xnx

) − 1

i log( xnx

)+O

(log x log2(Tx) +

log T

log x

),

where x > 1, T ≥ 2, and nx denotes the nearest prime power to x. This precise forms of Landau’s formulamaybe helpful in understanding of the geometric patterns appeared in this paper.

Acknowledgment

I greatly indebted to Professor A. Akbary for pointing out a big number of grammatical mistakes, andfor clearing mathematical content of the paper by asking some important questions. Also, I gratefullyacknowledge the many helpful suggestions of Professor J.-M. Deshouillers during the preparation of thepaper, and specially his suggestion for adding Figure 10, and introducing me the paper [14]. Finally, Ideem my duty to thank Professor R. Heath-Brown for giving very valuable comments on the mathematicaljustification of the geometric patterns described in this paper.

References

[1] A. Akbary, and M.R. Murty, Uniform distribution of zeros of Dirichlet series, Anatomy of integers. CRMProc. Lecture Notes, Vol. 46, pp. 143–158, Amer. Math. Soc. Providence RI, 2008.

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Figure 14: The space curves of the Weyl sums∑

n≤1000 e(n/e) (leftmost graph), and∑

n≤10000 e(an) with

an = (50/5001)n2 (mid graph), and an = (1/125)n3/2 (rightmost graph)

[2] F. Dekking, and M. Mendes France, Uniform distribution modulo one: a geometrical viewpoint, Journalfur die Reine und Angewandte Mathematik, 329 (1981), pp. 143–153.

[3] J.-M. Deshouillers, Geometric aspect of Weyl sums. Elementary and Analytic Theory of Numbers, BanachCenter Publications, 17 (1985), pp. 75–82.

[4] K. Ford, and A. Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function,Journal fur die Reine und Angewandte Mathematik, 579 (2005), pp. 145–158.

[5] A. Fujii, On the uniformity of the distribution of the zeros of the Riemann zeta function, Journal fur dieReine und Angewandte Mathematik, 302 (1978), pp. 167–205.

[6] A. Fujii, Some problems of Diophantine approximation in the theory of the Riemann zeta function. III,Commentarii Mathematici Universitatis Sancti Pauli, 42 (1993), pp. 161–187.

[7] M. Hassani, Geometric patters in uniform distribution of zeros of the Riemann zeta function, Mathe-matics Without Boundaries: Surveys in Pure Mathematics, pp. 245–258, Themistocles M. Rassias andPanos M. Pardalos (Eds.), Springer, New York, 2014.

[8] E. Hlawka, Uber die Gleichverteilung gewisser Folgen, welche mit den Nullstellen der Zetafuncktionenzusammenhangen, Osterr. Akad. Wiss., Math.-Naturw. Kl. Abt. II. 184 (1975), pp. 459–471.

[9] A. Ivic, The Riemann Zeta Function, John Wiley, 1985.

[10] L. Kuipers, and H. Niederreiter, Uniform distribution of sequences, Dover Publications, 2006.

[11] E. Landau, Uber die Nullstellen der ζ-Funktion, Mathematische Annalen, 71 (1911), pp. 548–568.

[12] H.L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and HarmonicAnalysis, CBMS, Vol. 84, Amer. Math. Soc., 1994.

[13] A.M. Odlyzko, Home Page Available at http://www.dtc.umn.edu/ odlyzko/, Tables of zeros of theRiemann zeta function.

[14] A.M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Mathematics ofComputation, 48 (1987), pp. 273–308.

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n≤10000 e(an) with an = n log n

[15] H. Rademacher, Fourier analysis in number theory, Symposium on harmonic analysis and relatedintegral transforms: Final technical report. Cornell Univ., Ithica. N.Y. 25 pages (1956); also in: H.Rademacher, Collected Works, pp. 434–458.

[16] G. Tenenbaum, and M. Mendes France, The prime numbers and their distribution, Student Mathemat-ical Library, Vol. 6, Amer. Math. Soc., 2001.

[17] E.C. Titchmarsh, The theory of the Riemann zeta function, 2nd ed., Revised by D.R. Heath-Brown,Oxford University Press, 1986.

[18] H. Weyl, Uber die Gleichverteilung von Zahlen mod. Eins., Mathematische Annalen, 77 (1916), pp.313–352.

86

The second dual of a Jordan triple system

A.A. Khosravi* and H.R. Ebrahimi Vishki

(Ferdowsi University of Mashhad, Mashhad, Iran)

Abstract: Mimicking the Arens idea, we extend the triple product of a Jordan triple system to the sec-ond dual. Then we compare the extended triple products with those triple products introduced by Dinnenbased on holomorphy theory.

keywords. Jordan triple system, Triple product, Second dual

subject. Primary: 17C65; Secondary 17A40, 46H70, 46K70, 46H25.

1 Introduction

Jordan triple systems were studied by Kaup [5] in his work on bounded symmetric domains. There aremany connections between Jordan triple systems and some other theories in mathematics. For instance,Tits-Kantor-Koecher construction theorem establishes a connection between Jordan triple systems and aclass of graded Lie algebras, (see [3]).

A Jordan triple is a real or complex Banach space M together with a continuous triple product ·, ·, · :M ×M ×M →M satisfying the following conditions:

1. ·, ·, · is symmetric and trilinear;

2. ·, ·, · obeys the Jordan identity:

a, b, x, y, z = a, b, x, y, z − x, b, a, y, z+ x, ya, b, z, (21)

or equivalently:

φ(a, b)(x, y, z) = φ(a, b)(x), y, z − x, φ(b, a)(y), z+ x, y, φ(a, b)z,

for all a, b, x, y, z ∈M , where φ(a, b)(x) := a, b, x.

Note that φ(a, b) is sometimes denoted by ab and operators of this form are called box operators. AHermitian Jordan triple is a complex vector space M with a triple product ·, ·, · which satisfies the Jordanidentity (21) and is linear and symmetric in the outer variables and conjugate linear in the middle one. Bya Jordan triple system, we mean a Jordan triple or a Hermitian Jordan triple. A subtriple of a Jordan triplesystem M is a subspace N of M which is closed with respect to the triple product; that is, x, y, z ∈ N forall x, y, z ∈ N .

Any Banach algebra M is a Jordan triple equipped with the canonical triple product x, y, z = 12(xyz+

zyx). If M is a ∗-Banach algebra then it is a Hermitian Jordan triple with the canonical triple productdefined by x, y, z = 1

2(xy∗z + zy∗x).

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A linear map f : M → M between two Jordan triple systems M and N is called a Jordan triplehomomorphism if it preserves the triple product.

A Jordan triple system M is called commutative if the box operators ab and xy commute for alla, b, x, y ∈M . It can be shown that M is commutative if and only if

a, b, x, y, z = a, b, x, y, z = a, b, x, y, z, (a, b, x, y, z ∈M).

lemma 1.1 (see [3, Lemmas 1.2.2, 1.2.3, 1.2.4]). The triple product ·, ·, · of a Jordan triple system satisfiesthe following identities for all x, y, z.

x, y, x, y, z = x, y, x, y, z,x, y, x, z, x = x, y, x, z, x,

x, y, x, z, x, y, x = x, y, x, z, x, y, x.

remark 1.2. It should be remarked that, if a triple map ·, ·, · is symmetric in the outer variables andobeys the there identities in Lemma 1.1, then it satisfies the Jordan identity (21), (see [7]).

definition 1.3. Let (M, ·, ·, ·) be a Hermitian Jordan triple which is also a Banach space with the norm‖ · ‖. Then M is called a JB∗-triple system if it satisfies the following conditions:

1. for each a ∈M , the operator φ(a, a) from M to M is a hermitian operator with non-negative spectrum;

2. ‖a, a, a‖ = ‖a‖3 for each a ∈M .

As an example, any C∗-algebra with the triple product x, y, z = 12(xy∗z+zy∗x) is a JB∗-triple system.

An example of a Jordan triple system which is not a JB∗-triple is `1(Z2) with the canonical triple product.

An element e in a JB∗-triple systemM is called a tripotent if e, e, e = e. If e is a tripotent, then Menjoys the Peirce decomposition M = M0(e) ⊕M1(e) ⊕M2(e), where Mk(e) = x ∈ M : φ(e, e)(x) = k

2xfor k = 0, 1, 2. The Peirce projections associated to e are given by P2(e) = Q2

e, P1(e) = 2(φ(e, e)−Q2e) and

P0(e) = I − 2φ(e, e) + Q2e, where Qe(x) = e, x, e, for each x ∈ M. The subspace M2(e) is a JB∗-algebra,

with Jordan product defined by xy = xey and the involution x] = exe. In this case e is the identity ofM2(e). It should be noted that a JB∗-triple system does not necessarily contain a tripotent. Barton, Kaupand Upmeier [2] showed that in a JB∗-triple system M , every complex extreme point of the closure of theunit ball of M is a (regular) tripotent. However, if M is a dual space, then by Krein-Milman theorem Mhas many tripotents.

2 The second dual of a Jordan triple system

By the ultrafilter version of the principle of local reflexivity, Dineen [4] showed that the second dual M∗∗

of a JB∗-triple system M is itself a JB∗-triple system. The principle of local reflexivity shows that M∗∗ canbe embedded via an isometry JU : M∗∗ →MU into certain ultrapower MU of M . Then ·, ·, ·u, given by

m,n, pu = w∗ − limUJU (m), JU (n), JU (p), (m,n, p ∈M∗∗)

provides an extended triple product on M∗∗. Barton and Timoney [1] showed that the extended tripleproduct is separately w∗-continuous.

Parallel to Dineen’s product, we introduce six, generally different, (Aron-Berner) extensions of the tripleproduct to the second dual M∗∗ of a Jordan triple system M, (see [6] and also [8]). However, M∗∗ isnot necessarily a Jordan triple system with each of the mentioned six extensions. We investigate someconditions under which M∗∗ is a Jordan triple system with one or some of the these extensions. Note thatthe ultrapower triple product · · · u is always symmetric but it does not satisfy the Jordan identity (21).However, if an Aron-Berner extension of the triple product is symmetric in the outer variables, then byLemma 1.1, it satisfies the Jordan identity. We thus have the following result.

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The second dual of a Jordan triple system

theorem 2.1. Let (M, ·, ·, ·) be a Jordan triple system. Then the second dual M∗∗ is a Jordan triplesystem under an Aron-Berner extension of ·, ·, · if and only if the required extension is symmetric in theouter variables.

We finally compare the Aron-Berner extensions of a triple product with those raising from ultrapowers.In this regard, we present the following example.

example 2.2. Let M = `1(N). Then M equipped with pointwise product is a commutative Banach ∗-algebra enjoying a bounded approximate identity. Further, M is a Jordan triple system in a canonical way.We show that M∗∗ is a Jordan triple system with all of Aron-Berner extensions, however, there exists anultrafilter U such that the ultrapower triple extension ·, ·, ·u does not satisfy the Jordan identity.

We finally present some unanswered questions on the second dual of certain Jordan triple systems.

References

[1] T. Barton and R.M. Timoney, weak∗-continuity of Jordan triple products and it’s applications, Math.Scand. 59 (1986), 177–191.

[2] R. Braun, W. Kaup, and H. Upmeier, A holomorphic characterization of Jordan C∗-algebras, Math. Z.161 (1978), 277-290.

[3] C.-H. Chu, Jordan structures in geometry and analysis, Vol. 190. Cambridge University Press, 2011.

[4] S. Dineen, The second dual of a JB∗ triple system, North-Holland Mathematics Studies. Vol. 125. North-Holland, (1986) 67-69.

[5] W. Kaup, Hermitian Jordan triple systems and the automorphisms of bounded symmetric domains, NonAssociative Algebra and Its Applications, Kluwer Acad. Publ. (1994), 204-214.

[6] A.A. Khosravi, H.R. Ebrahimi Vishki and A. Peralta, Aron–Berner extensions of triple maps with ap-plication to the bidual of JB-triple systems, preprint.

[7] O. Loos, Jordan Pairs, Lecture Notes in Math. 460 Springer-Verlag, 1975.

[8] M. Niazi, M.R. Miri and H.R. Ebrahimi Vishki, Ternary weak amenability of the bidual of a JB∗-triple,Banach J. Math. Anal. 11 (2017), 676–697.

89

Kawada-Ito theorem for locally compact quantum groups

F. Khosravi*(IPM, Tehran, Iran)

Abstract: We will generalize the concept of compact quantum hypergroup to compact quantum hyper-system. We will consider the condition that guarantee the existance of Haar satates on compact quantumhypersystems. Finally we will prove that all idempotent states on locally compact quantum groups ariseas Haar states of compact quantum subhypersystems in a canonical way. This is an appropriate version ofKawada-Ito theorem for locally compact quantum group.

keywords. Locally compact quantum groups, Idempotent states, quantum subgroups.

subject. 46L65, 46L30

1 Introduction

We will refer to [3, 4] for the theory of locally compact quantum groups. There exist three concepts ofthis theory with different languages. The first concept is based on a von Neumann algebra and consists aquadruple (L∞(G),∆G, ϕ, ψ) where L∞(G) is a von Neumann algebra, the comultiplication ∆G is a normal,unital, ∗-homomorphism which is coassociative, and ϕ and ψ are n.s.f left and right Haar weights respec-tively. We will denote the predual of L∞(G), the space of all normal functionals on L∞(G), by L1(G). Twoothe concepts of a locally compact quantum group G are based on C∗-algebras the reduced C∗-algebra willbe denoted by C0(G) and the universal C∗-algebra by Cu0 (G).Given a locally compact quantum group G, the comultiplications ∆G and ∆u

G induce Banach algebra struc-tures on L1(G) and Cu0 (G)∗ respectively and We shall identify L1(G) with a subspace of Cu0 (G)∗ and underthis identification L1(G) forms a two sided ideal in Cu0 (G)∗.Closed quantum subgroups were studied by [2] in details. There exists two definitions of closed quantumsubgroups based on von Neumann setting by S.Vaes or universal C∗-algebraic by S. L. Woronowicz. It isknown that closed quantum subgroups in the sense of Woronowicz are closed in the sense of Vaes. In thispaper we work with Woronowicz-closed subgroups.

definition 1.1. [2] Let G and H be locally compact quantum groups. Then H is called a Woronowicz-closedquantun subgroup of G if there exists a surjective ∗-homomorphism π : Cu0 (G)→ Cu0 (H) such that

(π ⊗ π) ∆uG = ∆u

H π.

A closed quantum subgroup H is compact quantum subgroup if Cu0 (H) is unital.The study of idempotent state on quantum groups has been an appealing subject in the last years. Idem-potent measures on locally compact groups are positive probability measures which are idempotent withrespect to the convolution. All idempotent states on groups are characterized as Haar states of compact

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Kawada-Ito theorem for locally compact quantum groups

subgroups which is well-known as Kawada-Ito theorem. It was showed by [5] that the result is not longertrue for quantum groups by giving us a counterexample of idempotent states on 8-dimensional Kac-Paljutkinquantum group which are not arising from Haar states of compact subgroups. After the work of A. Van Daeleand his collaborator on algebraic quantum hypergroup, U. Franz and A. Skalski claim that the appropriategeneralization version of Kawada-Ito theorem in the concept of quantum groups should be as follows: allidempotent states on locally compact quantum groups arise (in a canonical way) as Haar states on compactquantum subhypergroups. But they were able to prove their claim only for finite dimensional cases, i.e.idempotent states on finite quantum groups arise (in a canonical way) as the Haar states on finite quantumsubhypergroups.

2 Main results

definition 2.1. The triple (S,∆S , RS) is called a hypersystem structure if

1. S is an operator system,

2. ∆ : S → S ⊗min S is a linear, unital, completely positive (completely bounded) map which is co-associative, i.e. (id⊗∆S) ∆S = (∆S ⊗ id) ∆S ,

3. RS : S → S is a unital, anti-linear, completely positive map such that σ(RS ⊗ RS) ∆S = ∆S RSand R2

S = id.

Let S∗ denotes the set of all (completely) bounded linear functionals on S, Then S∗ is a unital Banach∗-algebra with the following product and involution:

ξ · η := (ξ ⊗ η) ∆S

ξ] := ξ RSPositive definite elements of a hypersystem (S,∆S , RS) are those x ∈ S such that ξ · ξ](x) ≥ 0 for all ξ ∈ S∗.definition 2.2. For a hypersystem structure (S,∆S , RS), h ∈ st(S) is called a Haar state if

(h⊗ id) ∆S = (id⊗ h) ∆S = 1Sh.

theorem 2.3. Let (S,∆S , RS) be a hypersystem structure. If the linear space spanned by the positivedefinite elements is dense in S. Then there exists a self-adjoint functional on S, which is left and rightinvariant.

Proof. Theorem 2.3 was proved for C∗-algebras in [1, Theorem 2.3] by the same technique. Let st(S), the setof all states on the operator system S, be denoted by Σ. Then Σ is closed with respect to the multiplicationand involution and is a compact set in the weak∗-topology. Let

L = Λ : Λ ⊆ Σ Σ · Λ ⊆ ΛE = Ξ : Ξ ⊆ Σ Ξ · Σ ⊆ Ξ

Consider L and E with the partial ordering induced by inclusion. A standard argument using Zorns lemmashows that there are minimal subsets Λ0 ∈ L,Ξ0 ∈ E. It is immediate that Ξ0 = λ∗ λ ∈ Λ0 . LetΩ = Λ0 ∩ Ξ0 The same strategy like [1, Theorem 2.3] yields Ω = Ξ0 · Λ0 = ν and for every λ ∈ Λ0,

ν · ν = ν, ν · λ = ν, λ∗ · ν = ν, λ∗ · λ = ν, λ · ν = λ. (22)

Let p ∈ S be a positive definite element, then consider p as a positive linear functional on ∗-algebra generatedby elements λ ∈ Λ0, λ

∗, ν. Using Cauchy-Schwarz inequality for the inner product and (22)

|(ν − λ)p|2 = |((ν − λ) · ν)p|2 ≤(((ν − λ)∗ · (ν − λ))p

)((ν · ν)p

)

where (ν − λ)∗ · (ν − λ) = ν · ν − λ∗ · ν − ν · λ + λ∗ · λ = 0. So for every positive definite element p ∈ S,ν(p) = λ(p) which by the density of these elements in S, we have λ = ν. Therefore Λ0 = Ξ0 = ν whichmeans that ν is a Haar state of (A,∆, R).

91

F. Khosravi

If the Haar state exists then it is unique since for Haar states h and h′ of S we have

h(s) = h(sh′(1S)) = h′(h(s)1S) = (h⊗ h′)∆(s) = h(h′(s)1S) = h′(s),

but it may not be faithful.

definition 2.4. A hypersystem structure (A,∆, R) is called compact quantum hypersystem if it admits afaithful Haar state.

definition 2.5. Let G be a locally compact quantum group and (S,∆S , RS ,hS) be a compact quantumhypersystem. Then (S,∆S , RS ,hS) is called a compact quantum subhypersystem of G, if there exists asurjective completely positive map πS : Cu0 (G)→ S such that

∆S πS = (πS ⊗ πS) ∆G.

definition 2.6. An idempotent state ω on a locally compact quantum group G will be said to arises as aHaar state on a compact quantum subhypersystem of G if there exists a compact quantum subhypersystem(S,∆S , RS ,hS), such that ω = hS πS .

The following theorem is the generalization of Kawada-Ito theorem for locally compact quantumgroup.

theorem 2.7. Let ω ∈ Cu0 (G)∗ be an idempotent state and p⊥ω ∈ Cu0 (G)∗∗ be it support projection. Thenthere exists a compact quantum subhypersystem (p⊥ωC

u0 (G)p⊥ω ,∆S , RS ,hS) of G, where

hS : p⊥ωCu0 (G)p⊥ω → C, hS(p⊥ω ap

⊥ω ) := ω(a).

References

[1] Y.A. Chapovsky, and L.I. Vainerman: Compact quantum hypergroups. J. of Operator Theorey, 41(1999), 261–289.

[2] M. Daws, P. Kasprzak, A. Skalski, and P. So ltan: Closed quantum subgroups of locally compact quantumgroups. Adv. Math., 231 (2012), 3473–3501.

[3] J. Kustermans, and S. Vaes: Locally compact quantum groups. Ann. Scient. Ec. Norm. Sup., 33 (2000),837–934.

[4] J. Kustermans, and S. Vaes: Locally compact quantum groups in the von Neumann algebraic setting.Math. Scand., 92 (2003), 68–92.

[5] A. Pal: A counterexample on idempotent states on a compact quantum group. Lett. Math. Phys., 37(1996), 75–77.

92

Best proximity points for weak MT -cyclic Reich

H. Lakzian* (Payame Noor University, Tehran, Iran)S. Barootkoob (University of Bojnord, Bojnord, Iran)

Abstract: In this paper, we introduce a notion of weakMT -cyclic Reich contractions with respect to aMT -function ϕ. Then we shall prove some new convergence and existence theorems of fixed point and bestproximity point for these contractions in uniformly Banach spaces. These results improve and generalizesome previous theorems in this field.

keywords. Cyclic map, best proximity point, weak MT -cyclic Reich contraction.

subject. 54H25, 47H10.

1 Introduction

The concept of cyclic contraction mappings in uniformly convex Banach spaces is defined by A. A. Eldredand P. Veeramani in [1]. They proved a theorem which is obtain a best proximity point for this contraction.This theorem ensures the existence of the best proximity point of cyclic contractions. For a cyclic mapT : A ∪ B → A ∪ B, Du and Lakian in [3] introduced a new class of maps called MT -cyclic contractionwith respect to ϕ on A ∪B which is contained the cyclic contraction maps as a subclass, see Example A in[3] for more information. Then they obtained some new existence and convergence theorems of iterates ofbest proximity points for these contractions. Many authors have investigated the existence, uniqueness andconvergence of iterates to the best proximity point under weaker assumptions over T . Afterward, Lakzianand Lin in [4] defined the concept of weak MT -cyclic Kannan contractions with respect to ϕ on A ∪ Bestablished some new convergence and existence theorems of the best proximity point theorems for thesecontractions in uniformly Banach spaces.

Let A and B be nonempty subsets of a nonempty set X. A map T : A ∪ B → A ∪ B is called a cyclicmap if T (A) ⊂ B and T (B) ⊂ A. Let (X, d) be a metric space and T : A ∪ B → A ∪ B a cyclic map. Forany nonempty subsets A and B of X, let

dist(A,B) = infd(x, y) : x ∈ A, y ∈ B.

A point x ∈ A ∪B is called to be a best proximity point for T if d(x, Tx) = dist(A,B).For c ∈ R, we recall that

lim supx→c

f(x) = infε>0

sup0<|x−c|<ε

f(x),

andlim supx→c+

f(x) = infε>0

sup0<x−c<ε

f(x).

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H. Lakzian, S. Barootkoob

definition 1.1. [2] A function ϕ : [0,∞) → [0, 1) is said to be a MT -function if it satisfies Mizoguchi-Takahashi’s condition, i.e. lim sups→t+0 ϕ(s) < 1 for all t ∈ [0,∞).

Obviously, if ϕ : [0,∞)→ [0, 1) is a nondecreasing or nonincreasing function, then ϕ is a MT -function.So, in particular, if ϕ : [0,∞)→ [0, 1) is defined by ϕ(t) = c, where c ∈ [0, 1), then ϕ is a MT -function.

Note that if ϕ is a MT -function then clearly ψ := 2ϕ3−ϕ is a MT -function.

example 1.2. [2] Let ϕ : [0,∞)→ [0, 1) be defined by

ϕ(t) :=

sin tt , if t ∈ (0, π2 ]0 , otherwise.

Since lim sups→0+

ϕ(s) = 1, ϕ is not an MT -function.

Recently, Du [2] proved some characterizations of MT -functions.

In this paper, we will introduce a notion of weak MT -cyclic Reich contractions with respect to a MT -function ϕ. Then we shall prove some new convergence and existence theorems for these contractions inuniformly Banach spaces. These results improve and generalize some previous theorems in this field.

2 Main Results

In this section, we present our main results. First we introduce the weak MT -cyclic Reich contraction withrespect to auxiliary MT -function ϕ.

definition 2.1. Let A and B be nonempty subsets of a metric space (X, d). Suppose that a map T :A ∪B → A ∪B satisfies

(MTR1) T (A) ⊂ B and T (B) ⊂ A;(MTR2) there exists a MT -function ϕ : [0,∞)→ [0, 1) such that

d(Tx, Ty) ≤ 1

3ϕ(d(x, y))[d(x, y) + d(x, Tx) + d(y, Ty)] + (1− ϕ(d(x, y)))dist(A,B),

for any x ∈ A and y ∈ B. Then T is called a weak MT -cyclic Reich contraction with respect to ϕ onA ∪ B. Also, T is called a weak MT -non − cyclic Reich contraction with respect to ϕ on A ∪ B if thecondition (MTR1) changes as follows:

(MTNR1) T (A) ⊂ A and T (B) ⊂ B;This contraction is said to be cyclic (resp. non-cyclic) Reich contraction, if ϕ ≡ α for some α ∈ [0, 1).

Note that in the above definition, clearly if A∩B 6= ∅, then dist(A,B) = 0 and T becomes the mappingfrom A ∩B into A ∩B and (MTR2) changes as follows:

d(Tx, Ty) ≤ 1

3ϕ(d(x, y))[d(x, y) + d(x, Tx) + d(y, Ty)].

There exists an example in [3] of a map T which is anMT -cyclic contraction but not a cyclic contraction.It is easy to see that the same example is also an MT -cyclic Reich contraction but it is not a cyclic Reichcontraction; so the class of MT -cyclic Reich contractions are bigger than their cyclic Reich contractions.

The discussion naturally splits into two cases of dist(A,B) 6= 0 and dist(A,B) = 0.In the case that dist(A,B) = 0, we can obtain the following theorem that generalize Reich theorem [5].

theorem 2.2. Suppose that A and B are nonempty closed subsets of a complete metric space (X, d) suchthat A∩B 6= ∅ and T : A∩B → A∩B is a weakMT -cyclic Reich contraction with respect to ϕ such thatfor any x ∈ A and y ∈ B,

d(Tx, Ty) ≤ 1

3ϕ(d(x, y))[d(x, y) + d(x, Tx) + d(y, Ty)]. (23)

Then T has a unique fixed point z in A ∩B.

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Best proximity points for weak MT -cyclic Reich

remark 2.3. Note that in the above theorem, if we delete the assumption A∩B 6= ∅, then we obtain a bestproximity point result in the uniformly convex Banach spaces; see Theorem 2.6 in below.

In the following theorem we prove a new existence theorem for weak MT -cyclic Reich contractions, forthe case of dist(A,B) 6= 0.

theorem 2.4. Let (X, d) be a metric space, let A and B be nonempty subsets of X. Let T : A∪B → A∪Bbe a weakMT -cyclic Reich contraction with respect to aMT -function ϕ. Let x ∈ A such that the sequenceT 2nx has a convergent subsequence in A. Then there exists a point z ∈ A such that d(z, Tz) = dist(A,B).

Taking ϕ ≡ α, where α ∈ [0, 1), in Theorem 2.4, we have the following corollary:

corollary 2.5. Let A and B be nonempty closed subsets of a complete metric space X. Let T : A∪B → A∪Bbe a Reich cyclic contraction map, x1 ∈ A and define xn+1 = Txn, n ∈ N. Suppose x2n−1 has a convergentsubsequence in A. Then there exists x ∈ A such that d(x, Tx) = dist(A,B).

For weakMT -cyclic Reich contractions, we prove the following convergence theorem, which is our mainresult in this section.

theorem 2.6. Let A and B be nonempty closed convex subsets of a uniformly convex Banach space. LetT : A ∪B → A ∪B be a weak MT -cyclic Reich contraction with respect to a MT -function ϕ. Then

(i) T has a unique best proximity point z in A,

(ii) The sequence T 2nx converges to z for any starting point x ∈ A,

(iii) z is the unique fixed point of T 2,

(iv) Tz is a best proximity point of T in B.

In the following corollary, we will obtain the best proximity point for Theorem Reich in [5] in uniformlyconvex Banach spaces.

corollary 2.7. Let A and B be nonempty closed convex subsets of a uniformly convex Banach space. LetT : A ∪B → A ∪B be a cyclic Reich contraction. Then

(i) T has a unique best proximity point z in A,

(ii) The sequence T 2nx converges to z for any starting point x ∈ A,

(iii) z is the unique fixed point of T 2,

(iv) Tz is a best proximity point of T in B.

Putting A ∩B 6= ∅ in Corollary 2.7, we get the following corollary as the special case.

corollary 2.8. Suppose that A and B are nonempty closed convex subsets of a uniformly convex Banachspace and T : A ∪B → A ∪B is a cyclic Reich contraction. Then T has a unique fixed point z in A ∩B.

95


References

[1] A. Anthony Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal.Appl. 323 (2006) pp. 1001-1006.

[2] W.-S. Du, On coincidence point and fixed point theorems for nonlinear multivalued maps, Topology andits Applications, 159 (2012) pp. 49-56.

[3] W.-S. Du, H. Lakzian, Nonlinear conditions and new inequalities for best proximity points, J. Inequalityand Applications 2012, 2012:206.

[4] I.-J. Lin, H. Lakzian, Best proximity point theorems for weak MT -cyclic Kannan contractions, Funda-mental Journal of Mathematics and Applications, 1 (1) (2018) pp. 1-9.

[5] S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull. 14 (1) 1971.

96

Topological center of generalized matrix Banach algebras

H. Lakzian* (Payame Noor University, Tehran, Iran)S. Barootkoob (University of Bojnord, Bojnord, Iran)

Abstract: In this paper, we will obtain the topological centers of a generalized matrix Banach algebraand then we are able to give new and large class of Banach algebras which are neither strongly irregular norregular; i.e. G 6= Z(G∗∗) 6= G∗∗ and G 6= Zt(G∗∗) 6= G∗∗ and Z(G∗∗) 6= Zt(G∗∗).

keywords. generalized matrix Banach algebra, topological center, Arens regular, strongly Arens irregular.

subject. 46H25, 46H20, 15A78

1 Introduction

Let A and B be two Banach algebras, M be an (A,B)-module and N be a (B,A)-module. Moreover,suppose that two bounded bilinear mappings Φ : M ×N → A and Ψ : N ×M → B are bimodule morphismson each of their coordinates which are satisfied in the following equalities:

m(nm′) = (mn)m′ and n(mn′) = (nm)n′ (n, n′ ∈ N,m,m′ ∈M);

Where mn := Φ(m,n) and nm = Ψ(n,m). Then G =[ A MN B

], with matrix-like addition and matrix-like

multiplication and norm given by

∥∥∥ a mn b

∥∥∥G

= ||a||+ ||m||+ ||b||+ ||n||,

is a Banach algebra which is said to be a generalized matrix Banach algebra.Generalized matrix algebras was first introduced by Sands in [5]. In the case where M = 0 or N =

0, G exactly degenerates to the triangular algebras. In general the generalized matrix algebra G is notisometrically isomorphic to the module extension Banach algebra (A+B) ⊕ (M+N), elsewhere MN = 0and NM = 0; in this case, these are called trivially generalized matrix algebras.

The aim of this paper is to compute the topological centers of a generalized matrix Banach algebra anduse this for obtaining a large class of Banach algebras which are neither strongly irregular nor regular; i.e.G 6= Z(G∗∗) 6= G∗∗ and G 6= Zt(G∗∗) 6= G∗∗ and Z(G∗∗) 6= Zt(G∗∗).

2 Topological center of Generalized Matrix Banach Algebras.

Let f : X×Y → Z be a bilinear mapping on normed spaces X,Y and Z and f∗∗∗ and f t∗∗∗t are the (Arens)extensions of f . Then the first and second topological centers of f is defined as follows.

Z(f) = x′′ ∈ X∗∗ : y′′ 7→ f∗∗∗(x′′, y′′) isweak∗ − weak∗ continuous,

97


Zt(f) = y′′ ∈ Y ∗∗ : x′′ 7→ f t∗∗∗t(x′′, y′′) isweak∗ − weak∗ continuous.f is said to be Arens regular if f∗∗∗ = f t∗∗∗t and strongly Arens irregular if Z(f) = X and Zt(f) = Y .

In the case where f is the product of a Banach algebra A, we denote f∗∗∗ and f t∗∗∗t by and ♦, and Z(f)and Zt(f) by Z(A∗∗) and Zt(A∗∗), respectively. We also say that A is Arens regular or strongly Arensirregular if f has the corresponding property. For more informations see [2].

In computing of the topological centers of G′′ we have four centers of Z(Φ), Zt(Φ), Z(Ψ) and Zt(Ψ),which are defined as follows.

Z(Ψ) = n′′ ∈ N∗∗; themap m′′ 7→ Ψ∗∗∗(n′′,m′′) : M∗∗ → B∗∗ isweak∗ − weak∗ continuous,Z(Φ) = m′′ ∈M∗∗; themap n′′ 7→ Φ∗∗∗(m′′, n′′) : N∗∗ → A∗∗ isweak∗ − weak∗ continuous,Zt(Ψ) = m′′ ∈M∗∗; themap n′′ 7→ Ψt∗∗∗(m′′, n′′) : N∗∗ → B∗∗ isweak∗ − weak∗ continuous,Zt(Φ) = n′′ ∈ N∗∗; themap m′′ 7→ Φt∗∗∗(n′′,m′′) : M∗∗ → A∗∗ isweak∗ − weak∗ continuous.

theorem 2.1. Suppose that A and B are two Banach algebras; M is a Banach (A,B)-bimodule and N is aBanach (B,A)-bimodule and Φ : M ⊗B N → A and Ψ : N ⊗AM → B are two bimodule homomorphisms.If there exist a′ ∈ A∗ and b′ ∈ B∗ such that Ψ∗(b′, N) = M∗, Ψ∗∗(M∗∗, b′) = N∗ and Φ∗(a′,M) = N∗,Φ∗∗(N∗∗, a′) = M∗, then Z(Ψ) = N and Z(Φ) = M .

lemma 2.2. For each[ α µν β

],[ α′ µ′

ν ′ β′]∈ G∗∗

[ α µν β

][ α′ µ′

ν ′ β′]

=[ αα′ + Φ∗∗∗(µ, ν ′) π∗∗∗` (α, µ′) + π∗∗∗r (µ, β′)π∗∗∗2 (ν, α′) + π∗∗∗1 (β, ν ′) Ψ∗∗∗(ν, µ′) + ββ′

],

and [ α µν β

]♦[ α µν β′

]=[ α♦α′ + Φt∗∗∗t(µ, ν ′) πt∗∗∗t` (α, µ′) + πt∗∗∗tr (µ, β′)πt∗∗∗t2 (ν, α′) + πt∗∗∗t1 (β, ν ′) Ψt∗∗∗t(ν, µ′) + β♦β′

],

where π`, πr are the left and right module actions on M and π1, π2 are the left and right module actions onN , respectively.

theorem 2.3. Suppose that G is a generalized matrix Banach algebra. Then

Z(G∗∗) =[ Z(A∗∗) ∩ Z(π`) Z(Φ) ∩ Z(πr)Z(π2) ∩ Z(Ψ) Z(B∗∗) ∩ Z(π1)

],

and

Zt(G∗∗) =[ Zt(A∗∗) ∩ Zt(π2) Zt(π`) ∩ Zt(Ψ)

Zt(π1) ∩ Zt(Φ) Zt(B∗∗) ∩ Zt(πr)],

Let us say that the Banach algebras A and B act regularly on the (A,B)−bimodule M if the moduleactions π` : A×M →M and πr : M ×B →M are regular. Then the following theorem is now immediate.

corollary 2.4. Let A and B be Banach algebras and M be a Banach (A,B)-bimodule and N be aBanach (B,A)-bimodule that act regularly on M and N . Then the generalized matrix Banach algebra

G =[ A MN B

]is Arens regular if and only if both A and B and also Φ and Ψ are Arens regular.

example 2.5. H = l1(N) with pointwise product is Arens regular. Define Φ : B(H) × H → H byΦ(T, x) = T (x), (T ∈ B(H), x ∈ H). Φ is not Arens regular since B(H) is infinite dimensional (See [3]).

Therefore G =[ H B(H)H H

]is not Arens regular, although the corner algebra H is Arens regular.

Taking N = 0 in Theorem 2.4, we obtain the following result.

corollary 2.6. ( Proposition 2.3 in [1]) Let A and B be Banach algebras which act regularly on a Banach

(A,B)-bimodule M . Then T =[ A M

0 B

]is Arens regular if and only if both A and B are Arens regular.

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Topological center of generalized matrix Banach algebras

remark 2.7. For Φ = 0 and Ψ = 0 in Theorem 2.4, the trivial generalized matrix Banach algebra G isArens regular if and only if both A and B are Arens regular.

Now we are able to give a large class of Banach algebras which are neither strongly irregular nor regular;i.e. G 6= Z(G∗∗) 6= G∗∗ and G 6= Zt(G∗∗) 6= G∗∗ and Z(G∗∗) 6= Zt(G∗∗).

theorem 2.8. Suppose that A and B are two strongly Arens irregular Banach algebras; M is a non-reflexive Banach (A,B)-bimodule with the zero left A−module action and N is a non-reflexive Banach(B,A)-bimodule with the zero right A−module action and Φ : M ⊗BN → A and Ψ : N ⊗AM → B are twobimodule homomorphisms such that there exist a′ ∈ A∗ such that Φ∗(a′,M) = N∗, Φ∗∗(N∗∗, a′) = M∗ and

Ψ = 0. Then G =[ A MN B

]is not left and right strongly Arens irregular. Also Z(G∗∗) 6= Zt(G∗∗).

remark 2.9. (i) For Φ = 0 and Ψ = 0 in Theorem 2.3, we obtain the topological center of moduleextension Banach algebra as in [4] page 249; also we have Theorem 5.1 in that reference.

(ii) Taking M = 0 or N = 0 in Theorem 2.3, we can conclude Theorem 6.1 in [4].

References

[1] B. E. Forrest, L. W. Marcoux, Weak Amenability of triangular Banach algrbras, Trans. Amer.Math. Soc. 345 (2002), 1435–1452.

[2] H. G. Dales, Banach Algebras and Automatic Continuity, vol. 24 of London Mathematical SocietyMonographs, The Clarendon Press, Oxford, UK, 2000.

[3] M. Gordji, Arens regularity of some bilinear maps, Proyecciones, 28 (1) (2009) 21–26.

[4] M. Eshaghi Gordji and M. Filali, Arens regularity of module actions, Studia Mathematica, 181(3) (2007) 237–254.

[5] A.D. Sands, Radicals and Morita contexts, J. Algebra 24 (1973) 335–345.

99

On convergence problem of Fourier series

M. Moazami Goodarzi*

(Shiraz University, Shiraz, Iran)

Abstract: In this talk we state a theorem on the (uniform) convergence of the Fourier series of functionsin the Schramm space ΦBV and outline its proof. This is an extension of a classical result of R. Salem.As a bi-product of the approach we take, a characterization of when the space ΛBV embeds into V [ν] isestablished; giving a necessary and sufficient condition that has been unknown for a long time.

keywords. Fourier series, Uniform convergence, Generalized bounded variation, Embedding.

subject. 42A20, 42A16, 46E35, 26A45.

1 Introduction

Jordan’s classical concept of bounded variation has proven fruitful in several areas, especially in the theoryof Fourier series where the Dirichlet–Jordan Theorem states that functions of bounded variation (BV) haveconvergent Fourier series. With the main motivation of extending this result to larger classes than BV,several authors have introduced various generalizations of this concept. We begin by recalling two of thesewhich are central to the talk.

Let Φ = φj∞j=1 be a sequence of increasing convex functions on the nonnegative reals such that φj(0) = 0for all j. We say that Φ is a Schramm sequence if 0 < φj+1(x) ≤ φj(x) for all j and

∑∞j=1 φj(x) =∞ for all

x > 0.

definition 1.1. A real-valued function f on the interval [a, b] is said to be of Φ-bounded variation ifVarΦ(f) = VarΦ(f ; [a, b]) := sup

∑nj=1 φj(|f(Ij)|) <∞ where the supremum is taken over all finite collections

Ijnj=1 of nonoverlapping subintervals of [a, b] and f(Ij) = f(sup Ij)− f(inf Ij).

We denote by ΦBV (= ΦBV[a, b]) the linear space of all functions f such that cf is of Φ-boundedvariation for some c > 0. Indeed, ΦBV turns into a Banach space—with a suitable norm—in which theHelly Selection Theorem holds (see [4]). Note that by making suitable choices of the sequence Φ, we mayobtain several function spaces which have been considered in the literature (see Remark 2.3 below).

definition 1.2. The modulus of variation of a function f on [a, b] is the nondecreasing concave sequenceνf (n) := sup

∑nj=1 |f(Ij)|, where the supremum is taken over all finite collections Ijnj=1 of nonoverlapping

subintervals of [a, b]. Any sequence ν with such properties is called a modulus of variation. The symbol V [ν]denotes the class of all functions f such that νf (n) = O(ν(n)) as n→∞.

100

On convergence problem of Fourier series

2 Main results

The subject of the talk is Theorem 2.1, which is the main result of [2] and gives a criterion for convergenceof the Fourier series of all functions of Φ-bounded variation. To prove this result, we will first establisha connection between the spaces ΦBV and V [ν] (Proposition 2.2), then will exploit it to infer the desiredconclusion from the celebrated result of Waterman on functions of harmonic bounded variation [5].

For each n, Φ−1n (x) denotes the inverse of the function Φn(x) :=

∑nj=1 φj(x), x ≥ 0. In the sequel, we

assume that [a, b] = [0, 1].

theorem 2.1. Let Φ be a Schramm sequence. If the condition

∞∑

n=1

Φ−1n (1)

n<∞ (24)

is fulfilled, then the Fourier series of any function f in ΦBV converges everywhere. Furthermore, thisconvergence is uniform over any closed interval of points of continuity of f .

The following result is of independent interest as it completely characterizes the embedding of ΦBV intoV [ν]. It is worth mentioning that in [3], among other results, the authors established a characterization ofwhen ΦBV embeds into generalized Wiener classes; ideas in whose proof are employed here. For a completeproof of Proposition 2.2, as well as other things, see [2].

proposition 2.2. Let Φ be a Schramm sequence and ν be a modulus of variation. Then ΦBV embeds intoV [ν] if and only if

lim supn→∞

nΦ−1n (1)

ν(n)<∞. (25)

Sketch of proof. The negation of (25) implies that there exists a sequence nk such that nk > 2k+2 andnkΦ

−1nk

(1)/ν(nk) > 24k. For each k, let mk be the greatest integer such that 2mk − 1 ≤ 2−knk, and definefk on [0, 1] in the following way:

fk(x) :=

2−kΦ−1nk

(1), x ∈ [2−k + 2j−2nk

, 2−k + 2j−1nk

); 1 ≤ j ≤ mk,

0, otherwise.

Since the fk have disjoint supports, f(x) :=∑∞

k=1 fk(x) is a well-defined function on [0, 1]. Then it can beshown that f ∈ ΦBV but f /∈ V [ν], which implies the necessity of (25).

To prove the sufficiency part, using the convexity of Φn along with an averaging trick [1], we show thatνf (n) ≤ (1 + VarΦ(f))nΦ−1

n

(1). Then (25) implies that νf (n) = O(ν(n)) as n→∞.

remark 2.3. Let Λ = λj∞j=1 be a Waterman sequence, i.e., a nondecreasing sequence of positive real

numbers such that∑∞

j=11λj

=∞ . If φj(x) := xλj

(j ≥ 1) in the above definition, the Waterman space ΛBV

is obtained. In particular, letting λj = j for all j we obtain the class HBV of functions of harmonic boundedvariation. Alternatively, if φ is an increasing convex function on the nonnegative reals such that φ(0) = 0,

limx→0+

φ(x)x = 0 and lim

x→+∞φ(x)x = +∞, then taking φj(x) := φ(x) for all j we get the class Vφ introduced by

Young. (See [5].)

corollary 2.4 ([2]). Let Λ be a Waterman sequence and ν be a modulus of variation. Then ΛBV embedsinto V [ν] if and only if

lim supn→∞

n

ν(n)∑n

j=11λj

<∞. (26)

Proof. As pointed out in Remark 2.3, if we take φj(x) = xλj

for j ≥ 1, then ΦBV = ΛBV. In that case, one

observes that Φ−1n (1) =

(∑nj=1

1λj

)−1, and thus (25) reduces to (26).

101

M. Moazami Goodarzi

Proof of Theorem 2.1. First, applying Proposition 2.2 with ν(n) := nΦ−1n (1), it is seen that ΦBV automat-

ically embeds into V [nΦ−1n (1)]. On the other hand, it is known [1, Theorem 2] that ΛBV contains V [ν]

whenever∑∞

n=1

(1λn− 1

λn+1

)ν(n) <∞. By means of this fact and letting λn := n (n ≥ 1), we conclude that

all functions of V [nΦ−1n (1)] lie in HBV provided

∞∑

n=1

( 1

n− 1

n+ 1

)nΦ−1

n (1) <∞.

But the latter condition is equivalent to (24). Consequently, the result follows by appealing to the HBVtheorem, which states that functions of HBV satisfy the Lebesgue Test [5].

remark 2.5. (i) Let φ be as in Remark 2.3. The complementary function of φ (in the sense of Young) isdefined to be ψ(x) := sup

y≥0xy − φ(y) for x ≥ 0. Salem proved (see [5]): If the condition

∞∑

n=1

ψ( 1

n

)<∞ (27)

holds, then the Fourier series of any continuous function f in Vφ converges uniformly. This was originallyproven by using the so-called Salem Test. Observe that when φj = φ for all j, (24) turns into the condition

∞∑

n=1

1

nφ−1

( 1

n

)<∞. (28)

However, it is a well-known fact that (27) and (28) are equivalent. So Theorem 2.1 extends Salem’s result.(ii) Relations between various function spaces arising from generalizations of bounded variation has been

investigated in several papers. For an account of this subject the reader is referred to [3]. We note thatby making spacial choices of Φ in Proposition 2.2, necessary and sufficient conditions for embedding theclasses φΛBV and Vφ into V [ν] are also obtained. In [1, Theorem 1], it was shown that V [n/(

∑nj=1 1/λj)]

contains ΛBV. However, in communication with Prof. F. Prus-Wisniowski, I learned that the problem ofcharacterizing when ΛBV embeds into V [ν] has been open; Corollary 2.4 answers this problem.

References

[1] M. Avdispahic, On the classes ΛBV and V [ν], Proc. Amer. Math. Soc. 95 (1985) 230–234.

[2] M. M. Goodarzi, A note on uniform convergence of Fourier series, preprint.

[3] M. M. Goodarzi, M. Hormozi, N. Memic, Relations between Schramm spaces and generalized Wienerclasses, J. Math. Anal. Appl. 450 (2017) 829–838.

[4] M. Schramm, Functions of Φ-bounded variation and Riemann–Stieltjes integration, Trans. Amer. Math.Soc. 267 (1985) 49–63.

[5] D. Waterman, The path to Λ-bounded variation, in: Recent advances in harmonic analysis and applica-tions in honor of Konstantin Oskolkov, D. Bilyk et al. (eds.), 2013, pp. 385–394.

102

A note on module generalized derivations of triangular Banach algebras

M. Momeni* and F. Kiany

(Islamic Azad University, ahvaz, Iran)

Abstract: In this paper we investigate bounded B−bimodule generalized derivation D : A → X on atriangular Banach algebras, where X is an A-B-bimodule.

keywords. module generalized derivation, Banach module, triangular Banach algebras

subject. 46H25

1 Introduction

The concept of module amenability for Banach algebras was introduced by Amini in [1]. Let B and A beBanach algebras such that A is a Banach B-bimodule with the following compatible actions;

α · (ab) = (α · a) b and (ab) · α = a (b · α) ,

for all a, b ∈ A, α ∈ B. Let X be a Banach A-bimodule and a Banach B-bimodule with compatibility ofactions;

α · (a · x) = (α · a) · x and a · (x · α) = (a · x) · α,

for all a ∈ A, α ∈ B, x ∈ X , and the same for the other side actions. Then we say that X is a BanachA-B-bimodule. If moreover, α · x = x · α, (α ∈ B, x ∈ X ), then X is called a commutative A-B-bimodule.

Let A and B be as above and X be a Banach A-B-bimodule. A bounded B-bimodule map D : A → Xis called a module derivation if

D (ab) = D (a) · b+ a ·D (b) , (a, b ∈ A) .

D is not necessarily linear, but its boundedness implies its norm continuity. Because it preserves sub-traction. When X is commutative A-B-bimodule, each x∈ X , defines a module derivation

Dx (a) = a · x− x · a, (a ∈ A) ,

which is called an inner module derivations.Let X be an A-B-bimodule. A linear mapping D : X → X is said to be a generalized derivation, if there

exists a derivation d : A → A such that

D (xa) = D (x) a+ xd (a) , (x ∈ X , a ∈ A) .

In this case, for convenience, we say that such a generalized derivation is a d-derivation. The conceptof generalized derivation was introduced and studied for the first time by Bresar in [2] and a number of

103

M. Momeni, F. Kiany

analysts have studied other properties of generalized derivations on Banach algebras, see [3,4].Now suppose that X is an A-B-bimodule. A continuous map T : A → X is called an B-bimodule map, iffor each a, b ∈ A and α ∈ A, T (a± b) = T (a)± T (b) and T (α · a) = α · T (a) and T (a · α) = T (a) ·α. Wesay that T : A → X is a left (right)-B-bimodule multiplier if it is an B-bimodule map and for each a, b ∈ A,

T (ab) = aT (b) (T (ab) = T (a) b ) .

Also T : A → X is an B-bimodule multiplier if it is a left and right multiplier and in this case we have,

T (ab) = aT (b) = T (a) b, (a, b ∈ A) .

The set of all left(right)-B-bimodule multipliers and B-bimodule multipliers is denoted by MLB (A,X )(Mr

B (A,X )) and MB (A,X ) Respectively.

2 Module generalized derivations of triangular Banach algebras

The main definitions of this paper is as follows;

definition 2.1. Let X be an A-B-bimodule and D : X → X be a bounded B-bimodule map. We say thatD is an B-bimodule generalized derivation if there exists an B-bimodule derivation d : A → A such that

D (x · a) = D (x) · a+ x · d (a) , (x ∈ X , a ∈ A) .

definition 2.2. Let X be an A-B-bimodule and D : A → X be a bounded B-bimodule map. We say thatD is an B-bimodule generalized derivation if there exists an B-bimodule derivation d : A → X such that

D (ab) = D (a) · b+ a · d (b) , (a, b ∈ A) .

In two above definitions, for convenience, we say that B-bimodule generalized derivation D is an B-bimodule d-derivation. Also in general the B-bimodule derivation d is not unique and it may happen thatd is not bounded. For instance if A acts on X trivially, i.e, X · A = 0 , then every bounded B-bimodulemap D is an B-bimodule d-derivation, for each derivation d on A.

definition 2.3. Let X be a commutative A-B-bimodule and D : A → X be an B-bimodule generalizedderivation. D is called B-bimodule generalized inner if there exists x ∈ X and T ∈ Mr

B (A,X ), such thatD (a) = T (a)− ax, (a ∈ A) .

Now let T =

[A XB

]=

[a x0 b

]: a ∈ A, b ∈ B, x ∈ X

be a triangular Banach algebra equipped

with the usual 2 × 2 matrix addition and formal multiplication. The norm on T is

∥∥∥∥[a x0 b

]∥∥∥∥ = ‖a‖A +

‖x‖X + ‖b‖B . Now, let A,B and X be Banach A-bimodules, and let X be a Banach (A,B)-bimodule (left

A-module and right B-module). We consider the Banach algebra I =

[α 00 α

]: α ∈ B

. The Banach

algebra T =

[A XB

]with usual 2 × 2 matrix product is an I-bimodule. In fact, that is isomorphic to

A⊕`1X⊕`1B as a Banach space and as a Banach A-bimodule.

proposition 2.4. Let X be a left Banach (A,B)-bimodule. Let DA : A → A be an B-bimodule dA-derivation and DB : B → B, DX : X → X be B-bimodule dB-derivations such that DX (ax) = aDX (x) +DA (a)x, (a ∈ A, x ∈ X ) . Then D : T → T with defination

D([

a x0 b

])=

[DA (a) DX (x)

0 DB (b)

]

, (a ∈ A, x ∈ X , b ∈ B) is an I-bimodule generalized derivation.

104

A note on module generalized derivations of triangular Banach algebras

proposition 2.5. LetA, B be unital Banach B-bimodules and X be an (A,B)-B-bimodule. If D : T → T bean I-bimodule generalized derivation, then there exists x0, x1 ∈ X , and B-bimodule generalized derivationsDA : A → A and DB : B → B and B-bimodule map DX : X → X , such that for each a ∈ A, b ∈ B andx ∈ X ;

1. D([

a 00 0

])=

[DA (a) ax0

0 0

],

2. D([

0 00 b

])=

[0 x1b0 DB (b)

],

3. D([

0 x0 0

])=

[0 DX (x)0 0

],

Moreover DX (ax) = DA (a)x+ aρ (x) and DX (xb) = DX (x) b+ xdB (a), (a ∈ A, x∈ X , b∈ B) .

example 2.6. Let X be a commutative Banach (A,B)-B-bimodule and T =

[A XB

]. Suppose that

x0 ∈ X be fixed. Define D1 : T → T with D1

[a x0 b

]=

[0 ax0 + x0b0 0

]and d1 : T → T with

d1

[a x0 b

]=

[0 ax0 − x0b0 0

], (a ∈ A, x∈ X , b∈ B) . Since X is a commutative, D1 and d1 are I-bimodule

map because for each a ∈ A, x∈ X , b∈ B and α ∈ A, we have

D1

([α 00 α

] [a x0 b

])= D1

([αa αx+ αb0 αb

])

=

[0 αax0 + x0αb0 0

]

=

[α 00 α

] [0 ax0 + x0b0 0

]

=

[α 00 α

]D1

[a x0 b

].

Similarly D1

([a x0 b

] [α 00 α

])= D1

[a x0 b

] [α 00 α

]. It is easy to see that d1 is an I -bimodule

map, too. Also,

D1

([a x0 b

] [a x0 b

])= D1

[aa ax+ xb0 bb

]=

[0 aax0 + x0bb0 0

].

On the other hand,

D1

([a x0 b

])[a x0 b

]+

[a x0 b

]d1

([a x0 b

])

=

[0 ax0 + x0b0 0

] [a x0 b

]+

[a x0 b

] [0 ax0 − x0b0 0

]

=

[0 ax0b+ x0bb0 0

]+

[0 aax0 − ax0b0 0

]

=

[0 aax0 + x0bb0 0

].

So D1 is an I-bimodule d1-derivation.

105

M. Momeni, F. Kiany

example 2.7. Let X be a Banach (A,B)-B-bimodule and T =

[A XB

]. Suppose that x0 ∈ X be fixed.

Define D2 : T → T with D2

[a x0 b

]=

[0 bx0 − x0 0

]and d2 : T → T with d2

[a x0 b

]=

[0 −x0 0

],

(a ∈ A, x∈ X , b∈ B) . If X · B = B · X , then D2 is an I-bimodule d2-derivation. Because,

D2

([a x0 b

] [a x0 b

])= D2

[aa ax+ xb0 bb

]=

[0 bbx0 − (ax+ xb)0 0

].

On the other hand,

D2

([a x0 b

])[a x0 b

]+

[a x0 b

]d2

([a x0 b

])

=

[0 bx0 − x0 0

] [a x0 b

]+

[a x0 b

] [0 −x0 0

]

=

[0 bx0b− xb0 0

]+

[0 −ax0 0

]

=

[0 bx0b− xb− ax0 0

].

And it is easy to see that D2 and d2 are I-bimodule maps. So D2 is an I-bimodule d2-derivation.

Acknowledgment

The author would like to thank the Ahvaz Branch, Islamic Azad University, Research Council for theirfinancial support.

References

[1] M. Amini, Module amenability for semigroup algebras, Semigroup Forum 69 (2004), 243-254.

[2] M. Bresar, On the distance of the compositions of two derivations to the generalized derivations, GlasgowMath. J. 33 (1991), 80-93.

[3] M. Bresar, A. R. Villena, The noncommutative Singer-Wermer conjecture and derivations, J.LondonMath. Soc. (2) 66 (2002), no.3, 710-720.

[4] B. Hvala, Generalized derivations in prime rings, Comm. Alg. 26(4) (1998), 1147-1166.

106

Charachterisation of p-frames in light of Hahn-Banach Theorem

S. Movahed*

(Sistan and Baluchestan University, Zahedan, Iran)

Abstract: In this paper we discuss equivalence of p-farmes in separable Banach spaces by analysis andsynthesis operator and give some charachterisation of this frames by use of a result of Hahn-Banach theoremand clarify the relationship between p-frames and complete sequence.

keywords. complete sequence, p-frame, q-Riesz basis


1 Introduction

Frames for Hilbert spaces were first introduced by Duffin and Schaeffer [3] in 1952 to study nonharmonicFourier series. Since then variouse generalization of frames have been developed by many authors. p-frameson of the important generalizations of frames in Banach spaces. Christencen and Steova[2] have deeplystudied p-frames in separable Banach spaces. In this paper we discuss some equivalence in p-frames. Let Xbe a separable Banach space with daual X∗. Suppose 1 < p, q < ∞, where 1

p + 1q = 1. A countable family

gi ⊂ X∗ is a p-frame for X if there exist constants A,B such that

A‖f‖ 6(∑

|gi(f)|p) 1p6 B‖f‖,∀f ∈ X

gi is a p-Bessel sequence if at least the upper p-frame condition is satisfied. Let Y be a Banach space. Afamily gi ⊂ Y is a q-Riesz basis for Y if spangi = Y and there exist constants A,B > 0 such that forall finite scalars sequences di,

A(∑

|di|q) 1q ≤ ‖

∑digi‖Y ≤ B

(∑|di|q

) 1q.

Note that completeness is part of our definition of a q-Riesz basis. Study of a p-frame gi ⊂ X∗ for X isbased on an analysis of the two operators

U : X → `p, Uf := gi(f), (29)

T : `q → X∗, Tdi :=∑

digi. (30)

U is called the analysis operator, and T is synthesis operator. Range of analysis operator U denoted byR(U).

107

S. Movahed

2 Main results

theorem 2.1 ([2]). Let gi ⊂ X∗ be a p-frame for X. Then the following are equivalent

1) gi is a q-Riesz basis for X∗,

2) If di ∈ `q and∑digi = 0, then di = 0,∀i,

3) R(U) = `p,

4) gi has a biorthogonal sequence fi ⊂ X, i.e., a family for which gj(fi) = δi,j ,

5) gi is minimal, i.e., for each j, gj /∈ spangii 6=j .

.

This result cannot be correct for a p-frame which is not a q-Riesz basis. on The other hand, p-framegi ⊂ X∗ is a q-Riesz basis if gi has a unique dual. A sequence xn in a normed space X is said to becomplete if spanxn = X. So by use of Hahn-Banach theorem, a sequence xn ⊂ X is complete if andonly if for n = 1, 2, ... and f ∈ X∗, f(xn) = 0 implies that f = 0.p-frame gi is a q-Riesz basis for X∗ if and only if R(U) = `p. This shows that if gi ⊂ X∗ is a p-frame forX, then analysis operator U is injective. If p-frame gi is a q-Riesz basis then U is surjective and synthesisoperator T is injective and so, on this case T and U are bijective. p-frame gi is minimal if and only if gihas a biorthogonal sequence fi ⊂ X, i.e., a family for which gj(fi) = δi,j(Kronecker, delta). Hahn-Banachtheorem shows that for a sequence xn, there exists a biorthogonal sequence yn if and only if xn isminimal. If xn is minimal, then the biorthogonal sequence yn is unique if and only if xn be complete.In this case we say that p-frame gi has a unique dual if and only if gi be a q-Riesz basis. A q-Rieszbasis gi for X∗ is by definition complete in X∗, so by [4] the biorthogonal sequence fi is unique and wecall it the dual of gi. A q-Riesz basis for X∗ is a special case of a p-frame for X.Theorem 2.2 shows that a p-frame has a dual if and only if R(U) is complete in `p.

theorem 2.2 ([2]). Let gi ⊂ X∗ be a p-frame for X, then the following are equivalent

1) R(U) is complemented in `p,

2) The operator U−1 : R(U)→ X can be extended to a bounded linear operator V : `p → X,

3) There exists a q-bessel sequence fi ⊂ X for X∗, such that f =∑∞

i=1 gi(f)fi, ∀f ∈ X.

References

[1] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, Boston, 2016.

[2] O. Christensen and D. Stoeva, p-frames in separable Banach spaces, Adv. Comput. Math, 18(2003),117-126.

[3] R. J. Duffin, A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Math. Soc. 72 (1952) 341-366.

[4] R.M.Young, An Introduction to Nonharmonic Fourier series, (Academic press, New York, 1980).

108

Some conditions of derivations of tensor products on nonassociativealgebras and some results about generalized derivations on C∗-algebras

R. Pirali* and M. Momeni

(Islamic Azad University Ahvaz Branch, khozestan , Iran)

Abstract: In the current article, we obtain the following results:Let R and S be nonassociative unital algebras. Suppose that either at least one of R and S is finite di-mensional or they both are finitely generated.Then we have some results for every derivation of R⊗ S. Byconsidering the triple product a, b, c, we introduce the study of linear maps which are triple derivationsor triple homomorphisms at a point.

keywords. Haar measure, Lebegue measure, Measure theory (between 3 to 5 keywords)


1 Introduction

Let R and S be nonassociative algebras. What are natural examples of derivations of the tensor productalgebra R ⊗ S? First of all, just as in any algebra, every element u from the nucleus gives rise to thederivation x → ux − xu. Next, given a derivation f of S and an element z from the center of R, the mapgiven by x ⊗ y → zx ⊗ f(y) is a derivation of R ⊗ S. Similarly, x ⊗ y → g(x) ⊗ wy defines a derivationof R ⊗ S for every derivation g of R and every central element w ∈ S . The goal of this short paper is toprove that under rather mild assumptions namely, both R and S are unital and either one of them is finitedimensional or both are finitely generated every derivation of R ⊗ S is the sum of derivations of the threetypes just described. From the nature of this result, and the relative simplicity of its proof, one would expectthat it is known; however, we have not been able to find it in the literature. Among related results, we firstmention the one by Block [[2], Theorem 7.1] which considers a similar situation, just that the assumptionthat R is unital is weakened and, on the other hand, S is assumed to be associative and commutative. Seealso [3] for some extensions of Blocks theorem. Benkart and Osborn dealt with the special case where Ris the (associative) matrix algebra Mn(f) [[4], Corollary 4.9]. Finally, in the case where both R and S areassociative, the description of derivations of R⊗ S can be (under some finiteness assumptions) obtained asa byproduct of results on Hochschild cohomology; see, for example, [[5], Corollary 3.4].Let A be a nonassociative (i.e., not necessarily associative) algebra over a field F . For x, y, z ∈ A we write

[x, y, z] = (xy)z − x(yz).

The setN(A) = n ∈ A|[n,A,A] = [A,n,A] = [A,A, n] = 0

is called the nucleus of A , and the set

Z(A) = z ∈ N(A)|zx = xz for all x ∈ A

109

R. Pirali, M. Momeni

is called the center of A. Of course, A is associative if and only if N(A) = A, and in this case the center issimply the set of elements that commute with all elements in A. We will consider the case where A = R⊗S,the tensor product of unital algebras R and S. It is therefore important to note that

N(R⊗ S) = N(R)⊗N(S)

, as one can readily check. Recall that a linear map d : A→ A is called a derivation if it satisfies

d(xy) = d(x)y + xd(y) for all x, y ∈ A.

By Der(A) we denote the set of all derivations of A. Further, for every u ∈ A we define Lu,Ru, adu : A→ Aby

Lu(x) = ux , Ru(x) = xu , adu = Lu−Ru. Note that adu ∈ Der(A) if u ∈ N(A). (If A is associative, such a derivation is said to be inner; innonassociative algebras one defines inner derivations somewhat differently, cf . [[6], p. 21]).

The following simple lemma will be needed in the proof of the main result .

lemma 1.1. Let R and S be nonassociative algebras, let d be a derivation of R ⊗ S, and let si|i ∈ I bea basis of S. Suppose that S is unital. Then for each i ∈ I there exists a derivation di of R such that forevery x ∈ R we have

d(x⊗ 1) =∑

i∈Idi(x)⊗ si (31)

and di(x) = 0 for all but finitely many i ∈ I. Furthermore, if R is finitely generated, then di = 0 for all butfinitely many i .

In general, there may be infinitely many nonzero derivations di of R such that for each x ∈ R we havedi(x) = 0 for all but finitely many i. For example, this holds for the partial derivatives ∂

∂Xion F [X1, X2, ...].

In such a case, given any elements wi ∈ Z(S) we have that

d =∑

i∈Idi ⊗ Lwi

is a derivation of R⊗ S.Following the terminology set by J. Alaminos, M. Bresar, J. Extremera, and A. Villena in [ [7], 4] and

J. Li and Z. Pan in [8], we shall say that a linear operator G from a Banach algebra A into a BanachA-bimodule X is a generalized derivation if there exists ξ ∈ X∗∗ satisfying

G(ab) = G(a)b+ aG(b)− aξb (a, b ∈ A)

Every derivation is a generalized derivation, however there exist generalized derivations which are not deriva-tion .

Let X be a Banach A-bimodule over a Banach algebra A. A linear map δ : A → X is a derivationwhenever it is continuous and a derivation at an element which is left (or right) invertible(see [9]).J. Zhu,Ch. Xiong, and P. Li prove in [[10]] a significant result showing that, for a Hilbert space H, a linear mapδ : B(H)→ B(H) is a derivation if and only if it is a derivation at a non-zero point in B(H). It is furthershown that a linear map which is a derivation at zero need not be a derivation.

We shall introduce a new point of view by exploiting those properties of a C∗-algebra A which are relatedto the ternary product defined by

a, b, c =1

2(ab∗c+ cb∗a) (a, b, c ∈ A) (32)

A linear map T between C∗-algebras preserving the previous triple product is called a triple homomor-phism. A triple derivation on a C-algebra A is a linear map δ : A→ A satisfying the generalized Leibnitzsrule

δa, b, c = δ(a), b, c+ a, δ(b), c+ a, b, δ(c)

110

Some conditions of derivations of tensor products on nonassociative algebras and some results aboutgeneralized derivations on C∗-algebras

, for all a, b, c ∈ A. We recall that a *-derivation on a C∗-algebra A is a derivation D : A → A satisfyingD(a)∗ = D(a∗) for all a ∈ A . Examples of derivations on A can be given by fixing z ∈ A and definingDz : A → A as the linear map given by Dz(a) = [z, a] = za − az. It is known that every -derivation on aC∗-algebra is a triple derivation in the above sense. It is further known the existence of derivations on Awhich are not triple derivations.

definition 1.2. Let T : A→ A be a linear map on a C∗-algebra, and let z be an element in A. We shall saythat T is a triple derivation at z if z = a, b, c in A implies that T (z) = T (a), b, c+a, T (b), c+a, b, T (c)

definition 1.3. Let T : A → B be a linear map between C*-algebras, and let z be an element in A. Weshall say that T is a triple homomorphism at z if z = a, b, c in A implies that T (a), T (b), T (c) = T (z).

Let T be a continuous linear map on a unital C*-algebra. In Theorem 2.5 we prove that T being a triplederivation at the unit implies that T is a generalized derivation.

lemma 1.4. Let T : A→ B be a triple derivation at the unit of A. Then the following statements hold:(a) T (1)∗ = −T (1);(b) The identity T (p) = T (p)p+ pT (p)− pT (1)p, holds for every projection p in A

2 Main results

2.1 Nonassociative complex algebra

theorem 2.1. Let R and S be nonassociative unital algebras. Suppose that either at least one of R and Sis finite dimensional or they both are finitely generated. Then every derivation d of R⊗S can be written as

d = adu+

p∑

j=1

Lzj ⊗ fj +

q∑

i=1

gi ⊗ Lwi

where u ∈ N(R)⊗N(S), zj ∈ Z(R), wi ∈ Z(S), fj ∈ Der(S), andgi ∈ Der(R).

corollary 2.2. Let R and S be as in Theorem 2.1 . If every derivation of R is of the form adm for somem ∈ N(R) and every derivation of S is of the form adn for some n ∈ N(S), then every derivation of R⊗ Sis of the form adu for some u ∈ N(R)⊗N(S)

Proof. If g = adm, m ∈ N(R), and w ∈ Z(S), then g ⊗ Lw = ad(m ⊗ w). Similarly, if z ∈ Z(R) andf = adn, n ∈ N(S), then Lz ⊗ f = ad(z ⊗ n).

If R and S are associative, this corollary gets a simpler form: if both R and S have the property thatall their derivations are inner, then so does R ⊗ S. It would be interesting to find out whether or not thisalso holds without the finiteness assumptions.

Since the center of the matrix algebra Mn(F ) consists of scalar multiples of the identity matrix, andsince every derivation of Mn(F ) is, as is well-known, inner, the following result by Benkart and Osbornfollows immediately.

corollary 2.3 ([4], Corollary 4.9). Let S be an arbitrary nonassociative unital algebra. Then every deriva-tion d of Mn(S) can be written as d = adu+ f# where u ∈ Mn(N(S)) and f# is a derivation obtained byapplying a derivation f of S to each matrix entry.

corollary 2.4. Let S be an arbitrary nonassociative unital algebra. Then every derivation d of Tn(S) canbe written as d = adu+ f# where u ∈ Tn(N(S)) and f# is a derivation obtained by applying a derivationfof S to each matrix entry.

111

R. Pirali, M. Momeni

2.2 Generalized derivation

theorem 2.5. Let T : A → B be a continuous linear map which is a triple derivation at the unit of A.Then T is a generalized derivation.

Proof. Let us take a ∈ Asa. Since, for each tinR, eita is a unitary element in A and 1 = eita, 1, e−ita, wededuce that

T (1) = T (eita), 1, e−ita+ eita, T (1), e−ita+ eita, 1, T (e−ita). Taking the first derivative in t we get

0 = T (aeita), 1, e−ita − T (eita), 1, ae−ita+ aeita, T (1), e−ita− eita, T (1), ae−ita+ aeita, 1, T (e−ita) − eita, 1, T (ae−ita)

(33)

for every t ∈ R . Taking a new derivative at t = 0 in the last equality, we get

0 = 2T (a2), 1, 1 − 4T (a), 1, a+ 2T (1), 1, a2+ 2a2, T (1), 1− 2a, T (1), a, (34)

or equivalently,

2T (a2) = 2T (a)a+ 2aT (a) + 2aT (1)∗a− T (1)a2 − a2T (1)− T (1)∗a2 − a2T (1)∗.

Lemma 1.4(a) assures that T (1)∗ = −T (1), which implies that

T (a2) = T (a)a+ aT (a)− aT (1)a, (35)

for every a in Asa. Finally, let us take a, b, c ∈ Asawith ab = 0 = bc. Write b = b+ − b− with b+b− = 0 andbσ ≥ 0 for all σ ∈ ±. Find dσ ≥ 0 in A such that (dσ)2 = bσ(σ = ±). It is not hard to check (for example,by applying the orthogonality of the corresponding range projections) that adσ = dσc = 0 forσ = ±. Nowapplying (35) we get

aT (b)c = aT (b+)c− aT (b−)c

= a(T (d+)d+ +d+ T (d+))c− a(T (d−)d−+d− T (d−))c = 0.(36)

We deduce from [3, Theorem 2.11] that T is a generalized derivation.

References

[1] H. G. Dales: Banach Algebras and Automatic Continuity. London Math. Soc. Monographs,New Series,24, Oxford University Press, New York, 2000

[2] R.E. Block, Determination of the differentiably simple rings with a minimal ideal, Ann. of Math. 90(1969) 433459 .

[3] S. Azam, Derivations of tensor product algebras, Comm. Algebra 36 (2008) 905927

[4] G.M. Benkart, J.M. Osborn, Derivations and automorphisms of nonassociative matrix algebras, Trans.Amer. Math. Soc. 263 (1981) 411430

[5] J. Le, G. Zhou, On the Hochschild cohomology ring of tensor products of algebras, J. Pure Appl. Algebra218 (2014) 14631477

[6] R.D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, 1966.

[7] J. Alaminos, M. Bresar, J. Extremera, A. Villena, Maps preserving zero products, Studia Math. 193 (2)(2009) 131159

112

Some conditions of derivations of tensor products on nonassociative algebras and some results aboutgeneralized derivations on C∗-algebras

[8] J. Li, Z. Pan, Annihilator-preserving maps, multipliers, and derivations, Linear Algebra Appl. 423(2010)513

[9] F. Lu, Characterizations of derivations and Jordan derivations on Banach algebras, Linear Algebra Appl.430 (89) (2009) 22332239.

[10] J. Zhu, Ch. Xiong, P. Li, Characterizations of all-derivable points in B(H), Linear Multilinear Algebra64 (8) (2016) 14611473.

[11] W. Kaup, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces,Math. Z. 183 (1983) 503529.

[12] S. Ayupov, K. Kudaybergenov, A.M. Peralta, A survey on local and 2-local derivations on C- and vonNeuman algebras, in: Contemp. Math., vol. 672, Amer. Math. Soc., 2016, pp. 73126.

113

Perturbation of woven-weaving fusion frames in Hilbert spaces

A. Rahimi* and Z. Samadzadeh


Abstract: A new notion in frame theory has been introduced recently under the name of woven-weaving frames by Bemros et. al. In this paper, Motivating by the concepts of fusion and weaving frames,we introduce the notion of woven-weaving fusion frames and present some of their features. It is well knownthat the perturbation theory is a paramount component in the study of frames. In this paper, we show thatthose of fusion frames that are small perturbations of each other, constitute woven-weaving fusion frames.

keywords. Woven-Weaving Frames, Fusion Frames, Woven-Weaving Fusion Frame, Perturbation

subject. 42C15, 42C30, 42C40

1 Introduction

Frames for Hilbert space were first introduced by Duffin and Schaeffer [7] to study non-harmonic Fourierseries in 1952. After some decades, Daubechies, Grossmann and Meyer reintroduced frames with extensivestudies, in 1986 [6] and popularized frames from then on. Such that, in the past thirty years, the frametheory became an attractive research and powerful tool for studies in signal processing, image processing,data compression, sampling theory and etc.

In one of the direction of applications of frames in signal processing, a new concept of woven-weavingframes in a separable Hilbert space introduced by Bemrose et. al. [1, 4]. In 2003, new type of frameswere presented to the scientific community, with name of frame of subspaces, by Cassaza and Kutyniok [2],which are now known as fusion frames. Improving and extending the notions of fusion and woven (weaving)frames, we investigate the new notion under the name woven (weaving) fusion frames. Also, we study somenew results conserning the consepts fusion and woven under perturbation.

2 Frames and Woven Frames in Hilbert Space

definition 2.1. A countable sequence of elements fii∈I in H is a frame for H, if there exist constants0 < A,B <∞ such that:

A‖f‖2 ≤∑

i∈I| 〈f, fi〉 |2 ≤ B‖f‖2, ∀f ∈ H. (37)

The numbers A and B are celled frame bounds. The frame fii∈I is called tight frame, if A = B andis called Parseval frame if A = B = 1. Also the sequence fii∈I is called Bessel sequence, if the upperinequality in (54) holds.

For m ∈ N, let [m] = 1, 2, ...,m :

114


definition 2.2. Let F = fiji∈I for j ∈ [m] be a family of frames for the separable Hilbert space H. Ifthere exist universal constants C and D, such that for every partition σjj∈[m] of I and for every j ∈ [m],

the family Fj = fiji∈σj is a frame for H with bounds C and D, then F is said a woven frames. For every

j ∈ [m], the frames Fj = fiji∈σj are said weaving frames. The constants C and D are called the lower

and upper woven frame bounds.

2.1 Fusion and Woven Fusion Frames

definition 2.3. Let νii∈I be a family of weights such that νi > 0 for all i ∈ I. A family of closed subspacesWii∈I of a Hilbert space H is called a fusion frame (or frame of subspaces) for H with respect to weightsνii∈I, if there exist constants C,D > 0 such that:

C‖f‖2 ≤∑

i∈Iν2i ‖PWi(f)‖2 ≤ D‖f‖2, ∀f ∈ H, (38)

where PWi is the orthogonal projection of H onto Wi. The constants C and D are called the lower and upperfusion frame bounds, respectively. If the second inequality in (57) holds, the family of subspace Wii∈I iscalled a Bessel sequence of subspaces with respect to νii∈I with Bessel bound D.

Extending and improving the notions of fusion and weaving frames, we introduce the notion of wovenfusion frames.

definition 2.4. A family of fusion frames Wij∞i=1, for j ∈ [m] , with respect to weights νiji∈I,j∈[m], issaid woven fusion frames if there are universal constant A and B, such that for every partition σjj∈[m] ofI , the family Wiji∈σj ,j∈[m] is a fusion frame for H with lower and upper frame bounds A and B. Eachfamily Wiji∈σj ,j∈[m] is called a weaving fusion frame.

For abrivation, we use W.F.F instead of the statement of woven fusion frame. The following theoremstates the equivalence conditions between woven frames and woven fusion frames (W.F.F).

theorem 2.5. Suppose for every i ∈ I, Ji is a subset of the index set I and νi, µi > 0. Let fi,jj∈Ji andgi,jj∈Ji be frame sequences in H with frame bounds (Afi ,Bfi) and (Agi ,Bgi) respectively. Define

Wi = span fi,jj∈Ji , Vi = span gi,jj∈Ji , ∀i ∈ I,

and choose orthonormal bases ei,jj∈Ji and e′i,jj∈Ji for each subspaces Wi and Vi, respectively. Supposethat

0 < Af = infi∈IAfi ≤ Bf = sup

i∈IBgi <∞

and0 < Ag = inf

i∈IAfi ≤ Bg = sup

i∈IBgi <∞.

Then the following conditions are equivalent:

(i) νifi,ji∈I,j∈Ji and µigi,ji∈I,j∈Ji are woven frames in H.

(ii) νiei,ji∈I,j∈Ji andµie′i,j

i∈I,j∈Ji

are woven frames in H.

(iii) Wii∈I and Vii∈I are W.F.F in H with respect to weights νii∈I , µii∈I, respectively.

example 2.6. Suppose ek∞k=1 be an orthonormal basis of H and H = `2 (N). For every i ∈ N, Hi =span ek∞k=i and eij∞j=1 = ei+j−1∞j=1 is an orthonormal basis for Hi. Let Pi∞i=1 and P ′i∞i=1 be the

family of orthogonal projections Pi : H −→ span ei and P ′i : H −→ span ei, ei+1 for each i ∈ N. Also,let fi,j = Pi(ei,j) and gi,j = P ′i (ei,j). Then we have:

fi,j = Pi (ei+j−1) =

ei j = 10 j > 1

, gi,j = P ′i (ei+j−1) =

ei j = 1ei+1 j = 20 j > 2

,

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A. Rahimi, Z. Samadzadeh

so fi,j∞i,j=1 is a tight frame with bound A = B = 1 and gi,j∞i,j=1 is a frame with bounds A = 1 and B = 2,such that these frames constitute woven frames. Because for arbitrary set σ ⊂ N and for every f ∈ H, wehave

‖f‖2 ≤∑

i∈σ

∞∑

j=1

|〈f, fi,j〉|2 +∑

i∈σc

∞∑

j=1

|〈f, gi,j〉|2 ≤ 2∞∑

i=1

|〈f, ei〉|2 = 2‖f‖2.

Now, if for every i ∈ N, we assume the set Ji = N, Wi = span fi,jj∈Ji and Vi = span gi,jj∈Ji . By Theorem

2.5 , Wi∞i=1 and Vi∞i=1 constitute W.F.F with weights νi = µi = 1(∀i ∈ N).

3 Main Results in Perturbation of Woven Fusion Frames

It is well known that the perturbation theory is a paramount component in the study of frames. In thissection, we show that those of fusion frames that are small perturbations of each other, constitute W.F.F.We start this section with Paley-Wiener perturbation of weaving fusion frames and continue two results ofperturbations in the sequel.

theorem 3.1. Let Wii∈I and Vii∈I be fusion frames for H with weights νii∈I and µii∈I and fusionframe bounds (AW ,BW ) and (AV ,BV ), respectively. If there exist constants 0 < λ1, λ2, µ < 1 such that:

2

AW

(√BW +

√BV)(

λ1

√BW + λ2

√BV + µ

)≤ 1

and

‖TW,ν(f)− TV,µ(f)‖ ≤ λ1‖TW,ν(f)‖+ λ2‖TV,µ(f)‖+ µ. (39)

Then Wii∈I and Vii∈I are W.F.F .

theorem 3.2. Let Wii∈I and Vii∈I be fusion frames for H with weights νii∈I and µii∈I and fusionframe bounds (AW ,BW ) and (AV ,BV ), respectively and the operators (TW,ν , UW,ν) and (TV,µ, UV,µ) arethe synthesis and analysis operators for these frames. If there exist constants 0 < λ, µ, γ < 1, such thatλBW + µBµ + γ

√BW < AW and for f ∈ H and arbitrary σ ⊂ I, we have

‖T σW,νUσW,ν(f)− T σV,µUσV,µ(f)‖ ≤ λ‖T σW,νUσW,ν(f)‖+ µ‖T σV,µUσV µ(f)‖+ γ‖UσW,ν(f)‖.

Such that T σW,ν , UσW,ν , T σV,µ and UσV,µ are the same as Theorem 3.1. Then Wii∈I and Vii∈I are W.F.F,with universal woven bounds

(AW − λBW − µBV − γ

√BW

),(AW + λBW + µBV + γ

√BW

).

theorem 3.3. Let Wii∈I and Vii∈I be fusion frames for H with weights νii∈I and fusion frame bounds(AW ,BW ) and (AV ,BV ), respectively. Also, if there exist a constant K > 0, such that for every σ ⊂ I:

∑

i∈σν2i ‖PWi(f)− PVi(f)‖ ≤ Kmin

∑

i∈σν2i ‖PWi(f)‖,

∑

i∈σν2i ‖PVi(f)‖

,

then Wii∈I and Vii∈I are W.F.F.

References

[1] T. Bemrose, P. G. Casazza, K. Grochenig, M. C. Lammers, R. G. Lynch, Weaving Frames, J. Oper.Matrices, 10 (2016), pp. 1093-1116.

[2] P. G. Casazza, G. Kutyniok, Frames of Subspaces, Contemp Math. Amer. Math. Soc, 345 (2004), pp.87-113.

116


[3] P. G. Casazza, G. Kutyniok, Sh. Li, Fusion frames and distributed processing, Appl. Comput. Harmon.Anal, 25 (2008), pp. 114-132.

[4] P. G. Casazza, R. G. Lynch, Weaving properties of Hilbert space frames, J. Proc. SampTA, (2015),110-114.

[5] O. Christensen, An introduction to frames and Riesz Basis, Birkhauser, Boston, 2016.

[6] I. Daubechies, A. Grossmann, Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys, 27 (1986),pp. 1271-1283.

[7] R. J. Duffin, A. C. Schaeffer, A class of nonharmonic Fourier series, J. Trans. Amer. Math. Soc, 72(1952), pp. 341-366.

[8] A. Rahimi, Z. Samadzadeh, B. Daraby, Frame related operators for woven frames, International Journalof Wavelets and Multiresolution and Information processing (to appear).

[9] L. K. Vashisht and Deepshikha, Weaving K-frames in Hilbert spaces, Adv. Pure Appl. Math.arXiv:1710.09562v4.

[10] L. K. Vashisht, Deepshikha, S. Garg and G. Verma , On weaving fusion frames for Hilbert spaces, In:Proceedings of SampTA, (2017), pp. 381-385.

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Cyclic amenability of Lau product of Banach algebras

M. Ramezanpour*

(Damghan University, Damghan, Iran)

Abstract: In this note we give general nessesary and sufficent conditions for Lau product of Banachalgebras to be cyclic amenable. In particular, we extend some results on this topic.

keywords. Banach algebra, cyclic amenability, Lau product

subject. 46H05, 47B47, 43A20, 46H99

1 Introduction

Let A and B be Banach algebras and θ ∈ σ(B), the set of all nonzero multiplicative linear functionals onB. The θ-Lau product A ×θ B is a Banach algebra which is defined as the vector space A × B equippedwith the algebra multiplication

(a1, b1)(a2, b2) = (a1a2 + θ(b2)a1 + θ(b1)a2, b1b2) (a1, a2 ∈ A, b1, b2 ∈ B),

and the norm ‖(a, b)‖ = ‖a‖+‖b‖. This type of product was introduced by Lau [6] for certain class of Banachalgebras known as Lau algebras and was extended by Sangani Monfared [8] for the arbitrary Banach algebras.Recall that a Lau algebra is a Banach algebra which is the predual of a von Neumann algebra for which theidentity of the dual is a multiplicative linear functional. The study of this large class of Banach algebrasoriginated with a paper published in 1983 by Lau [6] in which he referred to them as F-algebras. Later on,in his useful monograph Pier introduced the name Lau algebra. Examples of Lau algebras include the groupalgebra and the measure algebra of a locally compact group or hypergroup, and also the Fourier algebra andthe Fourier-Stieltjes algebra of a topological group. We note that, the unitization A] of A can be regardedas the ι-Lau product A×ι C where ι ∈ σ(C) is the identity map.

This product provides not only new examples of Banach algebras by themselves, but also it can serve as asource of (counter) examples for various purposes in functional and harmonic analysis. From the homologicalalgebra point of view A×θ B is a strongly splitting Banach algebra extension of B by A, which means that,A is a closed two sided ideal of A×θ B and the quotient (A×θ B)/A is isometrically isomorphic to B. Thisextension enjoys some properties that are not shared in general by arbitrary strongly splitting extensions.For instance, commutativity does not preserve by a general strongly splitting extension, however, A×θ B iscommutative if and only if both A and B are commutative.

Some aspects of A×θB such as many basic properties, some notions of amenability and some homologicalproperties are investigated by many authors; see for example [1, 5] and [8]. In particular, It was shown in [1,Theorem 5.1] that if 〈A2〉 = A then the cyclic amenability of A×θ B is equivalent to the cyclic amenablilityof both A and B. In this note, we give general nessesary and sufficent conditions for A ×θ B to be cyclicamenable. In particular, we improve and extend [1, Theorem 5.1].

118


2 Cyclic amenability

Let A be a Banach algebra, and X be a Banach A-bimodule. Then the dual space X∗ of X becomes a dualBanach A-bimodule with the module actions defined by

(fa)(x) = f(ax), (af)(x) = f(xa),

for all a ∈ A, x ∈ X and f ∈ X∗. In particular, A∗ is a Banach A-bimodule. A derivation from A into X isa linear mapping D : A→ X satisfying

D(ab) = D(a)b+ aD(b) (a, b ∈ A).

If x ∈ X then dx : A → X defined by dx(a) = ax − xa is a derivation. A derivation D is inner if there isx ∈ X such that D = dx.

It is often useful to restrict one’s attention to derivations D : A→ A∗ satisfying the property

D(a)(c) +D(c)(a) = 0 (a, c ∈ A).

Such derivations are called cyclic. Clearly inner derivations are cyclic. A Banach algebra is called cyclicamenable if every continuous cyclic derivations D : A → A∗ is inner. This notion was presented byGronbaek [3]. He investigated the hereditary properties of this concept, found some relations between cyclicamenability of a Banach algebra and the trace extension property of its ideals. Examples of cyclic amenableBanach algebras include C∗-algebras, `1(G) if G is a group, `1(S) if S is the free semigroup on a set X and`1(N). If X is a Banach space with dimX > 1, then X with zero algebra product is an example of a Banachalgebra which is not cyclic amenable; see [3] for more details.

Ghahramani and Loy [2] introduced several approximate notions of amenability by requiring that allbounded derivations from a given Banach algebra A into certain Banach A-bimodules to be approximatelyinner. In the same paper, the authors showed the distinction between each of these concepts and thecorresponding classical notions and investigated properties of algebras in each of these new classes. Motivatedby this notions, Esslamzadeh and Shojaee [9] defined the concept of approximate cyclic amenability forBanach algebras and investigated the hereditary properties for this new notion. A Banach algebra A iscalled approximately cyclic amenable, if every continuous cyclic derivation D : A → A∗ is approximatelyinner; i.e. there exists a net fα ⊆ A∗ such that D = limα dfα in the strong operator topology.

Let A and B be Banach algebras and θ be an element of σ(B). The Banach space (A ×θ B)∗ can beidentified with the Banach space A∗ × B∗ equipped with the maximum norm ‖(f, g)‖ = max‖f‖, ‖g‖ inthe natural way. We can find that the (A×θ B)-bimodule actions on (A×θ B)∗ are formulated as follows:

(f, g)(a, b) = (fa+ θ(b)f, gb+ f(a)θ) ,

(a, b)(f, g) = (af + θ(b)f, bg + f(a)θ) ,

for a ∈ A, b ∈ B, f ∈ A∗ and g ∈ B∗.To clarify the relation between cyclic amenability of A ×θ B and that of A and B we begin with the

following lemma which plays a key role in the sequel.

lemma 2.1. Let A and B be two Banach algebras and θ ∈ σ(B). A continuous cyclic derivation D :A×θ B → (A×θ B)∗ enjoys the presentation

D(a, b) = (D1(a) + S(b), D2(b)− S∗(a)),

for all a ∈ A and b ∈ B, where

1. D1 : A→ A∗ and D2 : B → B∗ are continuous cyclic derivations.

2. S : B → A∗ is a bounded linear operator satisfying S(bd) = θ(b)S(d)+θ(d)S(b) and aS(b) = S(b)a = 0for all a ∈ A and b, d ∈ B.

119

M. Ramezanpour

Moreover, D is (approximately) inner if and only if S = 0 and both D1 and D2 are (approximately) inner.

Recall that, a linear functional d : B → C is called a point derivation at θ ∈ σ(B), if

d(b1b2) = θ(b1)d(b2) + θ(b2)d(b1) (b1, b2 ∈ B).

proposition 2.2. Let A and B be two Banach algebras and θ ∈ σ(B). Then there is no non-zero boundedlinear operator S : B → A∗ satisfying S(bd) = θ(b)S(d) + θ(d)S(b) and aS(b) = S(b)a = 0 for all a ∈ A andb, d ∈ B if and only if either 〈A2〉 = A or every continuous point derivation at θ is zero.

In the next, we gives general necessary and sufficient conditions for A×θ B to be cyclic amenable. Thisresult extends [1, Theorem 5.1].

theorem 2.3. [7] Let A and B be two Banach algebras and θ ∈ σ(B). Then A ×θ B is (approximately)cyclic amenable if and only if

1. A and B are (approximately) cyclic amenable.

2. Either 〈A2〉 = A or every continuous point derivation at θ is zero.

We recall from [4] that B is called left (resp. right) θ-amenable if every continuous derivation from Binto X∗ is inner, for every Banach B-bimodule X with b · x = θ(b)x (resp. x · b = θ(b)x); (b ∈ B, x ∈ X).This notion of amenability is a generalization of the left amenability of a class of Banach algebras studiedby Lau in [6], known as Lau algebras. Example of left (resp. right) θ-amenable Banach algebras includeamenable Banach algebras and the Fourier algebra A(G) for a locally compact group G.

As an immediate consequence of Theorem 2.3, we have the next result.

theorem 2.4. Under each of the following conditions, the (approximate) cyclic amenability of A ×θ B isequivalent to the (approximate) cyclic amenability of both A and B.

1. 〈A2〉 is dense in A.

2. A has a bounded left or right approximate identity.

3. A is weakly amenable.

4. There is no non-zero continuous point derivation at θ.

5. B is left or right θ-amenable.

6. B is weakly amenable.

As an immediate consequence of Theorem 2.3, we have the next result, which was proved in [9]; see also[3].

corollary 2.5. Let A be a Banach algebra. Then A] is (approximately) cyclic amenable if and only if A is(approximately) cyclic amenable.

References

[1] E. Ghaderi, R. Nasr-Isfahani, and M. Nemati, Some notions of amenability for certain products of Banachalgebras, Colloq. Math., 130 no. 2 (2013), 147–157.

[2] F. Ghahramani, and R. J. Loy, Generalized notions of amenability, J. Funct. Anal., 208 no. 1 (2004),229–260.

[3] N. Grønbæk, Weak and cyclic amenability for noncommutative Banach algebras, Proc. Edinburgh Math.Soc., 35 no. 2 (1992), 315–328.

120


[4] E. Kaniuth and A.T.-M. Lau and J. Pym, On ϕ-amenability of Banach algebras, Math. Proc. Camb.Phil. Soc., 144, (2008), 85–96.

[5] A. R. Khoddami, and H. R. Ebrahimi Vishki, Biflatness and biprojectivity of Lau product of Banachalgebras, Bull. Iranian Math. Soc., 39 no. 3 (2013), 559–568.

[6] A. T.-M. Lau, Analysis on a class of Banach algebras with applications to harmonic analysis on locallycompact groups and semigroups, Fund. Math. , 118 no. 3 (1983), 161–175.

[7] M. Ramezanpour, More on cyclic amenability of certain product of Banach algebras, Submitted.

[8] M. Sangani Monfared, On certain products of Banach algebras with applications to harmonic analysis,Studia Math., 178 (3) (2007), 277–294.

[9] G. H. Esslamzadeh, and B. Shojaee, Approximate weak amenability of Banach algebras, Bull. Belg. Math.Soc. Simon Stevin, 18 no. 3 (2011), 415–429.

121

Duality of controlled frames

M. Rashidi-Kouchi*

(Islamic Azad University, Kahnooj Branch, Kahnooj, Iran)

Abstract: Controlled frames are important tools to improve the numerical efficiency of iterative algo-rithms for inverting the frame operator on Hilbert spaces. In this paper, we study controlled dual framesin Hilbert spaces and characterize these frames by using of the operator theory and canonical controlleddual. We show that a controlled dual frame need not be a dual frame. We investigate the relation betweenbounds of controlled frames and their related frames.

keywords. Frame, Controlled frame, Dual, Hilbert space.

subject. 42C15, 42C40

1 Introduction

Frames for Hilbert spaces were first introduced by Duffin and Schaeffer [3] in 1952 in order to investigate somedeep problems in non-harmonic Fourier series. They were reintroduced in 1986 by Daubechies, Grossman,and Meyer [2] and popularized from then on. Owing to the increasing number of applications, considerableattention and effort have been paid towards development of frame theory in recent years. For a comprehensivesurvey of frames, dual frames and related topics, we refer to [2].

The last two decades have seen tremendous activity in the development of frame theory and manygeneralizations of frames have come into existence which include bounded quasi-projectors, fusion frames,pseudo-frames, oblique frames, g-frames, continuous frames, K-frames, fractional calculus and Hilbert-Schmidt frames. Work along this line has been performed, recently, in order to add further further ingredientswithin there generalized frames. One of the outcomes is development of controlled frames [1]. These typeof frames have been introduced to improve the numerical efficiency of iterative algorithms for inverting theframe operator on abstract Hilbert spaces. The main advantage of these frames lies in the fact that theyretain all the advantages of standard frames but additionally they give a generalized way to check the framecondition while offering a numerical advantage in the sense of preconditioning.

Throughout this paper, GL(H) is the set of all bounded linear operators on H with a bounded inverse.

definition 1.1. Let C ∈ GL(H). A frame controlled by the operator C or C-controlled frame is a familyof vectors ψjj∈J ⊆ H, such that there exist two constants AC > 0 and BC <∞ satisfying

AC∥∥f∥∥2 ≤

∑

j∈J

⟨f, ψj

⟩⟨Cψj , f

⟩≤ BC

∥∥f∥∥2,

for every f ∈ H.

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Every frame is an I-controlled frame. Hence the controlled frames are generalizations of frames. Thecontrolled frame operator SC is defined by

SCf =∑

j∈J

⟨f, ψj

⟩Cψj , f ∈ H.

Here, we study the notion of controlled dual frames in Hilbert spaces and provide the general charac-terizations of these frames by using the machinery of the operators. We show that a controlled dual frameneed not be a dual frame. We also investigate the relation between bounds of controlled frames and theirrelated frames.

2 Main results

Controlled dual frames defined by author and el al. in [5]. First, we recall this definition.

definition 2.1. Let H be a Hilbert space and C ∈ GL(H). Suppose that ψj : j ∈ J ⊆ H is C-controlledframe and φj : j ∈ J ⊆ H is a Bessel sequence. Then φj : j ∈ J ⊆ H is said to be a C-controlled dualof ψj : j ∈ J ⊆ H if the following condition

f =∑

j∈J〈f, φj〉Cψj , (2.1)

holds for all f ∈ H.

Using Proposition 3.2 in [1] and its proof, it follows that the controlled frame operator SC is positive,self-adjoint and invertible. Also SC = CS, where Sf is the frame operator associated with the standardframe ψj : j ∈ J. Thus, the following reconstruction formula holds

f = SCS−1C f =

∑

j∈J

⟨S−1C f, ψj

⟩Cψj =

∑

j∈J

⟨f, S−1

C ψj⟩Cψj , for all f ∈ H. (2.2)

Let ψj = S−1C ψj , then the above equalities lead us to

f =∑

j∈J

⟨f, ψj

⟩Cψj , f ∈ H.

Hence, the sequenceψj : j ∈ J

is a C-controlled dual sequence of ψj : j ∈ J and such a sequence is

called a canonical C-controlled dual sequence. If C is self-adjoint, then SC = CS and S−1C C = CS−1

C whichfurther implies that

f =∑

j∈J〈f, ψj〉Cψj .

The synthesis operator TCΨ : `2 → H associated with C-controlled frame Ψ = ψj : j ∈ J can be definedas

TCΨ

(αjj∈J

)=∑

j∈JαjCψj . (2.3)

Since Cψj : j ∈ J ⊆ H is a frame for H, the synthesis operator TCΨ is well defined and bounded. Fur-thermore, we can represent C-controlled frame operator as

SC = TCΨT∗Ψ,

where TΨ is the synthesis operator associated with standard frame Ψ = ψj : j ∈ J ⊆ H.

In the following example, we show that a controlled dual frame need not be a dual frame and vice-versa.

123

M. Rashidi-Kouchi

example 2.2. Let ej∞j=1 be an orthonormal basis for Hilbert space H and consider the frame ψj∞j=1 =e1, e1, e2, e3, . . . . Consider the operator C : H → H given by

C(ej) =ej2, j = 1, 2, 3, . . . .

Then, we can easily verify that C ∈ GL(H) andψj∞j=1

= e1, e1, 2e2, 2e3, . . . is a canonical C-controlled

dual frame for ψj∞j=1, whereasψj∞j=1

does not form a standard dual frame for ψj∞j=1, as∑∞

j=1〈f, ψj〉ψj =2f 6= f , for any f ∈ H. On the otherhand, the sequence

φj∞j=1 =

e1

3,2e1

3, e2, e3, . . .

,

constitutes a dual frame for ψj∞j=1 but not a canonical C-controlled dual frame for ψj∞j=1; because∑∞j=1〈f, φj〉Cψj = f/2 6= f , for any f ∈ H.

Next theorem characterize all C-controlled duals of a C-controlled frame.

theorem 2.3. Let H be a Hilbert space and C ∈ GL(H). Suppose that ψj : j ∈ J is a C-controlled framefor H. Then φj : j ∈ J is a C-controlled dual of ψj : j ∈ J if and only if φj : j ∈ J = V ej : j ∈ Jwhere ej : j ∈ J is the standard orthonormal basis of `2 and V : `2 → H is a bounded operator such thatTCΨV

∗ = I.

In the following theorem, we characterize C-controlled dual of a C-controlled frame by using the canon-ical C-controlled dual.

theorem 2.4. Let H be a Hilbert space and C ∈ GL(H). Suppose that ψj : j ∈ J is a C-controlled framefor H. Then, φj : j ∈ J is a C-controlled dual of ψj : j ∈ J if and only if

φj = S−1C ψj + U∗ej , (j ∈ J),

where ej : j ∈ J is the standard orthonormal basis of `2 and U ∈ B(H, `2) with TCΨU = 0.

Let H be a Hilbert space and C ∈ GL(H). Also, let Ψ = ψj : j ∈ J be a C-controlled frame with theoptimal bounds AC and BC , respectively and let A and B be the optimal bounds of Ψ = ψj : j ∈ J as ausual frame. Since

∑

j∈J

⟨f, ψj

⟩⟨Cψj , f

⟩≤

∑

j∈J

∣∣⟨f, ψj⟩∣∣2

1/2∑

j∈J

∣∣⟨Cψj , f⟩∣∣2

1/2

≤√B∥∥f∥∥√B

∥∥C∥∥∥∥f

∥∥

= B∥∥C∥∥∥∥f

∥∥2,

and, thus we obtain BC ≤ B∥∥C∥∥. In addition, we have

AC∥∥f∥∥2 ≤

∑

j∈J

⟨f, ψj

⟩⟨Cψj , f

⟩≤

∑

j∈J

∣∣⟨f, ψj⟩∣∣2

1/2√B∥∥C∥∥∥∥f

∥∥.

Therefore, we have

A2C

B‖C‖2∥∥f∥∥2 ≤

∑

j∈J

∣∣⟨f, ψj⟩∣∣2 ,

hence, A2C ≤ AB‖C‖2.

124


References

[1] P. Balazs, J-P. Antoine and A. Grybos, Weighted and controlled frames, Int. J. Wavelets Multiresolut.Inf. Process., 8(1): (2010), 109–132.

[2] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, Boston, 2016.

[3] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27(1986),1271–1283.

[4] R.J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc.72(1952), 341–366.

[5] M. Rashidi-Kouchi, A. Rahimi and Firdous A. Shah, Duals and multipliers of controlled frames in Hilbertspaces, Int. J. Wavelets Multiresolut. Inf. Process., 16(5): (2018), 1–13.

125

Some notes on controlled frames in Hilbert C∗-modules

M. Rashidi-Kouchi*

(Islamic Azad University, Kahnooj Branch, Kahnooj, Iran)

Abstract: Hilbert C∗-modules are a generalization of Hilbert spaces by allowing the inner product totake values in a C∗-algebra rather than in the field of complex numbers.

In this paper, controlled frames with two operators in Hilbert C∗-modules invesitgated. This generalizecontrolled frames in Hilbert and Hilbert C∗-module. We show, in Hilbert C∗-module setting, controlledframe with two operators is classical frame. Finally controlled frames with two operators in Hilbert C∗-modules characterized and sufficient conditions for a frame to be a controlled frame presented.

keywords. Frame, Controlled frame, Hilbert C∗-module.

subject. 42C15, 46L08

1 Introduction

Frames for Hilbert spaces were first introduced in 1952 by Duffin and Schaeffer [3] for study of nonharmonicFourier series. They were reintroduced and development in 1986 by Daubechies, Grossmann and Meyer[2],and popularized from then on.

Hilbert C∗-module forms a wide category between Hilbert space and Banach space. Its structure wasfirst used by Kaplansky in 1952. It is an often used tool in operator theory and in operator algebra theory.It serves as a major class of examples in operator C∗-module theory.

The notions of frames in Hilbert C∗-modules were introduced and investigated by Frank and Larson [4].They defined the standard frames in Hilbert C∗-modules in 1999 and got a series of result for standard framesin finitely or countably generated Hilbert C∗-modules over unital C∗-algebras. Extending the results to thismore general framework is not a routine generalization, as there are essential differences between HilbertC∗-modules and Hilbert spaces. For example, any closed subspace in a Hilbert space has an orthogonalcomplement, but this fails in Hilbert C∗-module. Also there is no explicit analogue of the Riesz representationtheorem of continuous functionals in Hilbert C∗-modules.

Hilbert C∗-modules form a wide category between Hilbert spaces and Banach spaces. Hilbert C∗-modules are generalizations of Hilbert spaces by allowing the inner product to take values in a C∗-algebrarather than in the field of complex numbers and define as follows:

Let A be a C∗-algebra with involution ∗. An inner product A-module (or pre Hilbert A-module) isa complex linear space H which is a left A-module with an inner product map 〈., .〉 : H × H → A whichsatisfies the following properties:

1. 〈αf + βg, h〉 = α〈f, h〉+ β〈g, h〉 for all f, g, h ∈ H and α, β ∈ C;

2. 〈af, g〉 = a〈f, g〉 for all f, g ∈ H and a ∈ A;

126


3. 〈f, g〉 = 〈g, f〉∗ for all f, g ∈ H;

4. 〈f, f〉 ≥ 0 for all f ∈ H and 〈f, f〉 = 0 iff f = 0.

For f ∈ H, we define a norm on H by ‖f‖H = ‖〈f, f〉‖1/2A . If H is complete with this norm, it is called a(left) Hilbert C∗-module over A or a (left) Hilbert A-module.

An element a of a C∗-algebra A is positive if a∗ = a and its spectrum is a subset of positive realnumbers. In this case, we write a ≥ 0. It is easy to see that 〈f, f〉 ≥ 0 for every f ∈ H, hence we define|f | = 〈f, f〉1/2. If a, b ∈ A and 0 ≤ a ≤ b, then ‖a‖ ≤ ‖b‖. Thus norm preserves order for positive membersin C∗-algebras.

We call Z(A) = a ∈ A : ab = ba,∀b ∈ A, the center of A. If a ∈ Z(A), then a∗ ∈ Z(A), and if a is aninvertible element of Z(A), then a−1 ∈ Z(A), also if a is a positive element of Z(A), then a1/2 ∈ Z(A). LetHom∗A(H,K) denotes the set of all adjointable A-linear operators from H to K and GL(H,K) as the set ofall adjointable bounded linear operators with an adjointable bounded inverse, and similarly for GL(H). IfT ∈ GL(H) is positive, i.e. 〈Tf, f〉 ≥ 0 for all f ∈ H, then we denote that by T ∈ GL+(H).

Let

`2(A) =

aj ⊆ A :

∑

j∈Ja∗jaj converges in‖.‖

with inner product

〈aj, bj〉 =∑

j∈Ja∗jbj , aj, bj ∈ `2(A)

and

‖aj‖ :=√‖∑

a∗jaj‖,

it was shown that [23], `2(A) is Hilbert A-modules.

Note that in Hilbert C∗-modules the Cauchy-Schwartz inequality is valid.

Let f, g ∈ H, where H is a Hilbert C∗-module, then

‖〈f, g〉‖2 ≤ ‖〈f, f〉‖ × ‖〈g, g〉‖.

We are focusing in finitely and countably generated Hilbert C∗- modules over unital C∗-algebra A. A HilbertA-module H is finitely generated if there exists a finite set x1, x2, ..., xn ⊆ H such that every x ∈ H can beexpressed as x =

∑ni=1 aixi, ai ∈ A. A Hilbert A-module H is countably generated if there exits a countable

set of generators.

The notion of (standard) frames in Hilbert C∗-modules is first defined by Frank and Larson [4].

Let H be a Hilbert C∗-module, and J a set which is finite or countable, a system fjj∈J ⊆ H is calleda frame for H if there exist constants C,D > 0 such that

C〈f, f〉 ≤∑

j∈J〈f, fj〉〈fj , f〉 ≤ D〈f, f〉 (40)

for all f ∈ H. The constants C and D are called the frame bounds. If C = D it called a tight frame and inthe case C = D = 1 it called Parseval frame. It is called a Bessel sequence if the second inequality in (40)holds.

Unlike Banach spaces, it is known that every finitely generated or countably generated Hilbert C∗-modules admits a frame but this is not true for every Hilbert C∗-module.

Weighted frames and controlled frames in Hilbert space have been introduced to improve the numericalefficiency of iterative algorithms for inverting the frame operator on abstract Hilbert spaces [1]. Also theauthor and Rahimi defined and investigated controlled frames in Hilbert C∗-modules [5].

In this paper, we extend controlled frames with two operators for Hilbert C∗-modules. This generalizecontrolled frames in Hilbert and Hilbert C∗-module. Smilar to Hilbert spaces, we show controlled frameswith two operators in Hilbert C∗-modules are classical frame.

127

M. Rashidi-Kouchi

2 Main results

In this section, we define and characterize controlled frames with two operators in Hilbert C∗-modules.Then we show every controlled frames with invertible bounded operators in Hilbert C∗-module are classicalframes in Hilbert C∗-modules.

definition 2.1. Let H be a Hilbert C∗-module and T, T ′ ∈ GL(H). Let F = fj : j ∈ J be a sequence inHilbert C∗-module H. The sequence F is called a controlled frame by T and T ′ or (T, T ′)-controlled frameif there exists two constants 0 < C,D <∞ such that

C〈f, f〉 ≤∑

j∈J〈f, Tfj〉〈T ′fj , f〉 ≤ D〈f, f〉,

for all f ∈ H. We call F a Parseval (T, T ′)-controlled frame if C = D = 1. If only the right inequality holds,then we call F a (T, T ′)-controlled Bessel sequence.

We call the (T, T )-controlled Bessel sequence and (T, T )-controlled frame, T 2-controlled Bessel sequenceand T 2-controlled frame with bounds C,D.

Let F = fj : j ∈ J be a Bessel sequence of elements in Hilbert C∗-module H. We define a linearoperator UTF : H→ `2(A) as follows:

UTF f = 〈f, Tfj〉j∈J ,for all f ∈ H. If F is also a (T, T ′)-controlled frame for H, then it is a bounded linear operator and this isthe analysis operator of TF . The adjoint operator U∗TF : `2(A)→ H which is called the synthesis operatoris defined as follows:

U∗TF (ajj∈J) =∑

j∈JajTfj ,

for all ajj∈J ∈ `2(A).Controlled frame operator STT ′ on Hilbert C∗-module H for controlled frame F is defined by

STT ′f := U∗T ′FUTF (f) =∑

j∈J〈f, Tfj〉T ′fj ,

for all f ∈ H.It is easy to see that STT ′ is well defined and

CIdH ≤ STT ′ ≤ DIdH.

Hence STT ′ is a bounded, invertible, self-adjoint and positive linear operator. Therefore we have STT ′ =S∗TT ′ = ST ′T . The following theorem charecterize controlled frames with two operators in Hilbert C∗-modules.

theorem 2.2. Let H be a Hilbert C∗-module, T, T ′ ∈ GL(H) and F = fjj∈J be a sequence in H. ThenF is a (T, T ′)-controlled frame for H if and only if there exist constants C,D > 0 such that

C‖f‖2 ≤

∥∥∥∥∥∥∑

j∈J〈f, Tfj〉〈T ′fj , f〉

∥∥∥∥∥∥≤ D‖f‖2, f ∈ H. (41)

The following proposition shows every (T, T ′)-controlled is a frame in Hilbert C∗-modules. Also it givesa condition that every classical frame is a (T, T ′)-controlled frame.

proposition 2.3. Let H be a Hilbert C∗-module, T, T ′ ∈ GLH and F = fj : j ∈ J a sequence in H.Then the following statements hold:(i) If F is a (T, T ′)-controlled frame for H, then F is a frame for H.(ii) If F is a frame for H and T ′SFT ∗ is a positive operator, then F is a (T, T ′)-controlled frame for H.

128


The following proposition shows that any frame is a T 2-controlled frame and versa.

proposition 2.4. Let H be a Hilbert C∗-module, T ∈ GL(H) be self-adjoint and F = fj : j ∈ J asequence in H. The sequence F is a frame if and only if F is a T 2-controlled frame.

The following proposition gives sufficient condition for a frame to be a controlled frame.

proposition 2.5. Let H be a Hilbert C∗-module, F = fj : j ∈ J a frame in H and T, T ′ ∈ G+L(H),which commute with each other and commute with SF . Then F is a (T, T ′)-controlled frame.

References

[1] P. Balazs, J-P. Antoine and A. Grybos. Wighted and Controlled Frames. Int. J. Wavelets Multiresolut.Inf. Process., 8(1) (2010), 109–132.

[2] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27(1986), 127-1-1283 .

[3] R. J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc.72(1952), 341–366 .

[4] M. Frank and D.R. Larson, A module frame concept for Hilbert C∗-modules, Functional and HarmonicAnalysis of Wavelets, San Antonio, TX, January 1999, Contemp. Math. 247, Amer. Math. Soc., Provi-dence, RI 207-233, (2000).

[5] M. Rashidi-Kouchi and A. Rahimi, controlled frames in Hilbert C∗-modules, Int. J. Wavelets Multiresolut.Inf. Process, 15(4) (2017), 1-15.

129

On the representation frames and duality

A. Razghandi* and A. A. Arefijamaal

( Hakim Sabzevari University, Sabzevar, Iran)

Abstract: In this paper we consider a kind of continuous frames induced by a unitary representationon a locally compact group, and characterize their dual.

keywords. Dual frames, irreducible representation, representation frames.

subject. 42C15; 20C99

1 Introduction

Representation frames on a locally compact group G are at least as old as modern frame theory. In 1985,Grossmann, Morlet and Paul [3] by using irreducible representations of locally compact groups describesa method for constructing reproducing systems. In 1986 Daubechies, Grossman and Meyer [2] describedGabor systems and wavelets in terms of unitary representation of the Weyl- Heisenberg and affine grouprespectively. In 2000, Han and Larson [4] established many of properties of representation frames. In thispaper, we consider continuous representation frame based on locally compact groups. We also investigatetheir dual of the form of representation and finding out under which conditions these duals are canonicaldual. Before state our results, we review some of the standard concepts and properties on frames andrepresentation frames.

Let H be a separable Hilbert space and X a locally compact Hausdorff space endowed with a positiveRadon measure ν. A mapping F : X → H is called a continuous frame if the mapping x → 〈F (x), φ〉 ismeasurable for all φ ∈ H and there exist constants 0 < A,B < +∞ such that

A‖φ‖2 ≤∫

X| 〈F (x), φ〉 |2dν(x) ≤ B‖φ‖2, (φ ∈ H) . (42)

A continuous frame is said to be tight when A = B. The mapping F is called Bessel if the second inequalityin (42) holds. Suppose that F is Bessel, then the operator T : L2(X)→ H defined by

Tφ =

∫

Xφ(x)F (x)dν(x)

is a bounded linear operator; so called the synthesis operator. Also, its adjoint given by

(T ∗φ)(x) = 〈φ, F (x)〉 , x ∈ X,

is called the analysis operator of F . The continuous frame operator, which is invertible, positive as well asself adjoint, is defined by SF = TT ∗. It is useful to reconstruct the elements of H as

φ = S−1F SFφ =

∫

X〈φ, F (x)〉S−1F (x) dν(x), (φ ∈ H) . (43)

130

On the representation frames and duality

The frame S−1F is called the Canonical dual of F . In general, if a frame K satisfies

φ =

∫

X〈φ, F (x)〉K (x) dν(x), (φ ∈ H) , (44)

then K is called alternate dual frame of F. In fact SF,K := TKT∗F = I.

Let G be a locally compact topological group by the normalized left Haar measure mG and the modularfunction ∆G. A unitary representation of G is a group homomorphism from G into the group of unitaryoperators on some nonzero Hilbert space H. A representation (π,H) is called irreducible if the only closedinvariant subspaces of H are 0 and H.

By a representation frame we mean a continuous frame as π(g)ψg∈G with bounds Aψ and Bψ where(π,H) is a unitary representation on G and ψ ∈ H. Denote its frame operator by Sψ. A Bessel vector for πis a vector ψ ∈ H such that π(g)ψg∈G is Bessel.

Let π be a square integrable representation, i.e., it is irreducible and

A :=

ψ ∈ H :

∫

G|〈π(g)ψ,ψ〉|2 dmG <∞

6= 0.

Then there exists a unique self adjoint operator C with the dense domain A such that for all ψ, φ ∈ A thefollowing orthogonality relation holds

∫

G〈f1, π(g)ψ〉〈f2, π(g)φ〉dmG(g) = 〈Cφ,Cψ〉〈f1, f2〉, (f1, f2 ∈ H) . (45)

Elements in A are called admissible vectors and C is known as the Duflo Moore operator see [1]. If Gis unimodular, then C is a multiple of the identity. For any admissible vector ψ, the continuous wavelettransform on G is defined by

Wψ : H → L2(G), (Wψϕ) (g) = 〈π(g)ψ,ϕ〉 (ϕ ∈ H) .

2 Main results

In this section, we consider representation frames and carry out some results on their duality.

proposition 2.1. Let (π,H) be unitary representation on group G and g ∈ G. Then the following areequivalent:

1. π is irreducible.

2. Every Bessel family π(g)ψg∈G is a tight frame.

3. The continuous wavelet transform Wψ is an isometry, for every admissible vector ψ.

theorem 2.2. Let (π,H) be an irreducible representation frame on G, also let φ and ψ ∈ H generate Besselfamilies. Then

1. Sψ,φ = 〈Cφ,Cψ〉 I, that C is Duflo Moore operator.

2. ‖Sψ,φ‖2 ≤ ‖Sψ‖‖Sφ‖.

In particular, if ‖Sψ‖‖Sφ‖ < 1 then the representation frames generated by ψ, φ are not dual frames.

corollary 2.3. Let π be a square integrable representation on locally compact group G and C the associatedDuflo Moore operator. Then for ψ, ψ ∈ H we obtain

1. π(g)ψg∈G andπ(g)C−2ψ

g∈G are dual pairs, for all ψ ∈ H with ‖ψ‖ = 1.

2. π(g)ψg∈G and

π(g) C−2

〈ψ,ψ〉 ψ

g∈Gare dual pairs.

131

A. Razghandi, A. A. Arefijamaal

3. π(g)ψg∈G and π(g)ψ − π(g) C−2

〈ψ,ψ〉 ψg∈G are strongly disjoint, when π(g)ψg∈G and π(g)ψg∈Gare dual pairs.

In the rest, we consider some conditions under which a dual of a representation frame is its canonicaldual.

proposition 2.4. Let (π,H) be unitary representation on group G and g ∈ G. Then the following areequivalent:

1. π(g)φg∈G is canonical dual for representation frame π(g)ψg∈G.

2. 〈ψ, π(g)φ〉 = 〈φ, π(g)ψ〉.

3. 〈S−1ψ ψ, π(g)ψ〉 = 〈ψ, π(g)φ〉.

4. 〈φ, π(g)ψ〉 =⟨ψ, π(g)S−1

ψ ψ⟩.

References

[1] S. T. Ali, J. P. Antoine, and J. P. Gazeau, Coherent States, Wavelets and Their Generalizations, NewYork, Springer-Verlag, 2000.

[2] I. Daubechies, A. Grossmann, Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys., 27 (1986)1271-1283.

[3] A. Grossmann, J. Morlet, T. Paul, Transforms associated to square integrable group representations. I.General results, J. Math. Phys., 26 (1985), 2473-2479.

[4] D. Han, D. Larson, Frames bases and group representations, Mem. Amer. Math. Soc. 697, 2000.

[5] G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., John Wiley, 1999.

132

Jordan amenability of Lau product of Jordan Banach algebras

H. Sadeghi Nahrekhalaji*

(Islamic Azad University, Feredan Branch, Isfahan, Iran)

Abstract: In this paper, for two Jordan Banach algebras A and B and θ ∈ ∆(B), the character spaceof B, it is shown that A×θ B (the Lau product of A and B) is Jordan amenable if and only if both A andB are Jordan amenable.

keywords. Jordan Banach algebra, Jordan derivation, Jordan amenability.

subject. Primary 46H25; Secondary 46M05.

1 Introduction

A real (resp. complex) Jordan algebra is a (non-necessarily associative) algebra over the real (resp. complex)field which whose product satisfying in the following conditions

1. a b = b a;

2. (a b) a2 = a (b a2),

for all a, b ∈ A. This algebras were introduced by P. Jordan, J. von Neumann and E. Wigner in orderto improve the quantum mechanics formalism (see [2]). A normed Jordan algebra is a Jordan algebra Aequipped with a norm, ‖.‖, satisfying ‖a b‖ ≤ ‖a‖‖b‖(a, b ∈ A). A Jordan Banach algebra is a normedJordan algebra whose norm is complete.

Let A be a Jordan algebra. A Jordan A-module is a vector space X equipped with a couple of bilinearproducts (a, x) 7→ a x and (x, a) 7→ x a from A×X to X, satisfying:

a x = x a, a2 (x a) = (a2 x) a, (46)

and2((x a) b

) a+ x (a2 b) = 2(x a) (a b) + (x b) a2, (47)

for every a, b ∈ A and x ∈ X (see §II.5, P.82 of [1] for the basic facts and definition of Jordan modules).Let X be a Jordan A-module. Then X is a Jordan Banach A-module if X is a Banach space and there

is a constant M ≥ 0 such that ‖a x‖ ≤M‖a‖‖x‖. For example, every associative Banach A-bimodule overa Banach algebra A is a Jordan Banach A-module for the product a x = 1

2(ax + xa)(a ∈ A, x ∈ X). Thedual, A∗, of a Jordan Banach algebra A is a Jordan Banach A-module with respect to the product

(a f)(b) = f(a b) (a, b ∈ A, f ∈ A∗).

The aim of the present work is to study Jordan amenability of Lau product A×θ B for Jordan Banachalgebras A and B. Indeed, it is proved that A ×θ B is Jordan amenable if and only if both A and B areJordan amenable.

133

H. Sadeghi Nahrekhalaji

2 Jordan amenable Banach algebras

We commence this section with the following definition:

definition 2.1. Let X be a Jordan Banach module over a Jordan Banach algebra A. A Jordan derivationfrom A into X is a linear map D : A→ X satisfying:

D(a b) = D(a) b+ a D(b) (a, b ∈ A).

The set of continuous Jordan derivation from A to X denote by DJ(A,X). A continuous Jordan derivationD : A → X is called an inner derivation if there exist x1, ..., xn in X and a1, ..., an in A such that D =∑n

i=1 δxi,ai (i.e. D(a) =∑n

i=1

((xi a) ai − (ai a) xi

)(a ∈ A)). The set of continuous inner Jordan

derivation from A to X denote by InnJ(A,X).

definition 2.2. A Jordan Banach algebra A is called Jordan amenable if every continuous Jordan derivationD : A −→ X∗ for some Jordan Banach A-module X is inner.

proposition 2.3. Let A be a Jordan amenable, B be a Jordan Banach algebra and ϕ : A → B be acontinuous homomorphism with dense range. Then B is Jordan amenable.

Proof. Let D : B → X∗ be a Jordan derivation for some Jordan Banach B-module X. By the followingA-module actions, X is also a Jordan Banach A-bimodule:

x • a = x ϕ(a), a • x = ϕ(a) x (a ∈ A, x ∈ X).

Define DA : A → X∗ by DA(a) = D(ϕ(a)

). It is clear that DA is a Jordan derivation on A. From the

Jordan amenability of A it follows that there exist x∗1, ..., x∗n in X∗ and a1, ..., an in A such that

DA(a) =n∑

i=1

((x∗i • a) • ai − (ai • a) • x∗i

)(a ∈ A).

For every b ∈ B, there exists a net (aα)α in A such that limα ϕ(aα) = b. So,

D(b) = limαDA(aα) = lim

α

n∑

i=1

((x∗i • aα) • ai − (ai • aα) • x∗i

)

= limα

n∑

i=1

((x∗i ϕ(aα)) ϕ(ai)−

(ϕ(ai) ϕ(aα)

) x∗i

)

=n∑

i=1

((x∗i b) ϕ(ai)− (ϕ(ai) b) x∗i

).

This means that D is an inner derivation. Therefore B is Jordan amenable.

corollary 2.4. If A is a Jordan amenable Jordan Banach algebra and if I is a closed ideal in A, then A/Iis Jordan amenable.

The following Proposition is needed for the proof of the main result.

proposition 2.5. Let A be a Jordan Banach algebra, and let I be a closed ideal of A such that both I andA/I are Jordan amenable. Then A is Jordan amenable.

Proof. LetX be a Jordan Banach A-module, and letD : A→ X∗ be a Jordan derivation. ThenD |I : I → X∗

is a Jordan derivation. Since I is Jordan amenable, there are x∗1, ..., x∗n in X∗ and a1, ..., an in I such that

D |A (a) =n∑

i=1

((x∗i a) ai − (ai a) x∗i

)(a ∈ I).

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Jordan amenability of Lau product of Jordan Banach algebras

Let D = D −∑ni=1 δx∗i ,ai . Then D |I= 0, and define D : A/I → X∗ by D(a+ I) = D(a)(a ∈ A). Let

F = f ∈ X∗ : a f = f a = 0 for all a ∈ I,

andX0 = lina x+ y b : a, b ∈ I, x, y ∈ X.

Then F ∼=(X/X0

)∗is a dull Jordan Banach A/I-module by action define by (a + I) • f = a f and

f • (a+ I) = f a for all a ∈ A. Let a ∈ I and b ∈ A. Then a D(b) = D(a b)− A(a) b = 0. Similarly,D(b) a = 0. It follows that D(A/I) ⊂ F . From the Jordan amenability of A/I, there exist y∗1, ..., y

∗n in F

and b1, ..., bn in A such that

D(a+ I) =m∑

j=1

((y∗j • (a+ I)) • (bj + I)− (bj a+ I) • y∗j

)(a ∈ A).

Consequently,

D(a) =n∑

i=1

((x∗i a) ai − (ai a) x∗i

)

+m∑

j=1

((y∗j a) bj − (bj a) y∗j

).

Thus D is a Jordan derivation. Therefore A is Jordan amenable.

3 Main Results

Let A and B be Banach algebras with ∆(B) 6= ∅. Let θ ∈ ∆(B). Then the direct product A×B equippedwith the algebra multiplication

(a1, b1) · (a2, b2) = (a1a2 + θ(b2)a1 + θ(b1)a2, b1b2) (a1, a2 ∈ A, b1, b2 ∈ B),

and the l1-norm is a Banach algebra which is called the θ-Lau product of A and B and is denoted by A×θB.This type of product was introduced by Lau [3] for certain class of Banach algebras and was extended byMonfared [4] for the general case.

We note that the dual space (A×θ B)∗ can be identified with A∗ ×B∗, via

〈(f, g), (a, b)〉 = 〈a, f〉+ 〈b, g〉 (a ∈ A, f ∈ A∗, b ∈ B, g ∈ B∗).

Moreover, (A×θ B)∗ is a (A×θ B)-bimodule with the module operations given by

(f, g) · (a, b) =(f.a+ θ(b)f, f(a)θ + g.b

), (48)

and

(a, b) · (f, g) =(a.f + θ(b)f, f(a)θ + b.g

), (49)

for all a ∈ A, b ∈ B and f ∈ A∗, g ∈ B∗.

proposition 3.1. Let A and B be Jordan Banach algebras. Then A×θ B is a Jordan Banach algebra.

theorem 3.2. Let A and B be Jordan Banach algebras. Then A ×θ B is Jordan amenable if and only ifboth A and B are Jordan amenable.

Acknowledgment

The author would like to thank the Islamic Azad university of Fereydan Branch for its support.

135

H. Sadeghi Nahrekhalaji

References

[1] N. Jacobson, Structure and representation of Jordan algebras, Amer. Math. Soc. Colloq. Publication, vol39, 1968.

[2] P. Jordan, J. Von Neumann and E. Wigner, On an algebraic generalization of the quantum mechanicalformalism, Ann. of Math. 35 (1934), 2964.

[3] A. T. Lau, Analysis on a class of Banach algebras with applications to harmonic analysis on locallycompact groups and semigroups, Fund. Math., 118 (1983), 161-175.

[4] M. S. Monfared, On certain products of Banach algebras with application to harmonic analysis, StudiaMath., 178 (2007), 277-294.

136

Some properties of woven frames and their related operators

Z. Samadzadeh*


Abstract: A new notion in frame theory has been introduced recently under the name woven-weavingframes by Bemrose et. al. In the studying of frames, some operators like analysis, synthesis, Gram andframe operator play the central role. In this paper, for the first time, we introduce and define these operatorsfor woven-weaving frames and review some properties of them.

keywords. Analysis operator, Frame operator, Synthesis operator,Weaving frames, Woven frames

subject. 42C15, 42C40, 65T60

1 Introduction

A frame as well as an orthonormal basis allows each element in Hilbert space to be written as an infinitelinear combinations of the frame elements, so that unlike the bases conditions, the coefficients might notbe unique. Frames did not seem to develop much interest, until Daubechies, Grossmann and Meyer [6]introduced their studies in 1986. During the last thirty years, the frame theory has been growing rapidly,especially after that Daubechies et. al introduced their studies. Thereafter, the frame theory be used inmore applications in: Signal processing, image processing, sampling theory and several applications.

In one of the direction of applications of frames in signal processing, a new concept of woven-weavingframes in a separable Hilbert space introduced by Bemrose et. al. [1, 4]. From the perspective of itsintroducers, woven frames has potential applications in wireless sensor networks that require distributeddata and signal processing. By the concepts of weaving, we introduce related operators for weaving andwoven frames, and investigate properties of this operators.

2 Frames and Woven Frames in Hilbert Space

definition 2.1. A countable sequence of elements fii∈I in H is a frame for H, if there exist constants0 < A,B <∞ such that:

A‖f‖2 ≤∑

i∈I| 〈f, fi〉 |2 ≤ B‖f‖2, ∀f ∈ H. (50)

The numbers A and B are celled frame bounds. The frame fii∈I is called tight frame, if A = B andis called Parseval frame if A = B = 1. Also the sequence fii∈I is called Bessel sequence, if the upperinequality in (54) holds. For a frame in H, we define the mapping:

U : H −→ `2 (I) , U(f) = 〈f, fi〉i∈I .

137

Z. Samadzadeh

The operator U is usually called the analysis operator. The adjoint operator of U is given by:

T : `2(I) −→ H, T ci =∑

i∈Icifi,

and is called the synthesis operator. By composing U and T , we obtain the frame operator:

S : H −→ H, S(f) = TU(f) =∑

i∈I〈f, fi〉 fi.

The operator S is positive, self-adjoint and invertible and the familyS−1fi

i∈I is said the canonical dual

frame of fii∈I.

definition 2.2. Let F = fiji∈I for j ∈ [m] be a family of frames for the separable Hilbert space H. Ifthere exist universal constants C and D, such that for every partition σjj∈[m] of I and for every j ∈ [m],

the family Fj = fiji∈σj is a frame for H with bounds C and D, then F is said a woven frames. For every

j ∈ [m], the frames Fj = fiji∈σj are said a weaving frame. The constants C and D are called the lower

and upper woven frame bounds.

3 Related Operators for Woven and Weaving Frames

In this section, for the first time, we introduce analysis, synthesis and frame operators of weaving and wovenframes. As a prerequisite for this operators, we define the following space. For each family of subspaces(`2(I)

)j

j∈[m]

of `2(I), we have

(`2(I)

)j

=

ciji∈σj | cij ∈ C , σj ⊂ I,

∑

i∈σj|ci|2 <∞

, ∀j ∈ [m].

We define the space:

∑

j∈[m]

⊕(`2(I)

)j

`2

=ciji∈I,j∈[m] | ciji∈I ∈

(`2(I)

)j,∀j ∈ [m]

,

with the inner product ⟨ciji∈I,j∈[m] ,

c′iji∈I,j∈[m]

⟩=

∑

i∈I,j∈[m]

∣∣∣cijc′ij∣∣∣ ,

it is easy to show that this space is a Hilbert space.

theorem 3.1. The family fiji∈I,j∈[m] is a Bessel woven if and only if the operator

TF :

∑

j∈[m]

⊕(`2(I)

)j

`2

−→ H , TF ciji∈I,j∈[m] =∑

i∈I,j∈[m]

cijfij

is well defined, linear and bounded.

corollary 3.2. Suppose that the family fiji∈I,j∈[m] is a Bessel woven forH. Then, the series∑

i∈I,j∈[m] cijfij

converges unconditionally for all ciji∈I,j∈[m] ∈(∑

j∈[m]

⊕(`2(I)

)j

)`2

.

Like frames and its extensions, we can characterize a woven frame in term of its woven frame operator.

138

Some properties of woven frames and their related operators

definition 3.3. Let F = fiji∈I,j∈[m] be a woven Bessel. Then for every partition σjj∈[m], the family

Fj = fiji∈σj for j ∈ [m] is a Bessel sequence. Therefore, we define the analysis, synthesis and frame

operator of Fj by

Uσj : H −→(`2(I)

)j, Uσj (f) = 〈f, fij〉i∈σj , ∀j ∈ [m], f ∈ H,

Tσj :(`2(I)

)j−→ H, Tσj ciji =

∑

i∈σjcijfij , ∀ ciji ∈

(`2(I)

)j.

Sσjf = TσjUσjf = Tσj 〈f, fij〉i∈σj =∑

i∈σj〈f, fij〉 fij ,

The operator Sσj is bounded, self-adjoint and invertible. Now, we define the analysis and synthesis operatorsfor the Bessel woven F = fiji∈I,j∈[m]:

UF : H −→

∑

j∈[m]

⊕(`2(I)

)j

`2

, UF (f) = 〈f, fij〉i∈I,j∈[m] ,

and

TF :

∑

j∈[m]

⊕(`2(I)

)j

`2

−→ H, TF ciji∈I,j∈[m] =∑

i∈I,j∈[m]

cijfij .

Also, by combination of UF and TF , the woven frame operator SF , for all f ∈ H, is defined by

SF : H −→ H, SF f = TFUF f =∑

i∈I,j∈[m]

〈f, fij〉 fij .

The operator SF is bounded, linear and self-adjoint operator.

In the next theorem, we demonstrate that the woven frames are equivalent to boundedness of wovenframe operator.

theorem 3.4. Let fiji∈I,j∈[m] be finite family of Bessel sequences in H. Then the following conditionsare equivalent:

(i) fiji∈I,j∈[m] is woven frames with universal woven frame bounds C and D.

(ii) for the operator SF f =∑

i∈I,j∈[m] 〈f, fij〉 fij , we have CIH ≤ SF ≤ DIH.

The next result shows that we can constitute tight woven frames from every woven frames by weavingoperators.

theorem 3.5. Let F = fiji∈I,j∈[m] be woven frame for H with universal woven bounds C and D and the

woven frame operator SF . If we define the positive square root of S−1F with S

− 12

F , then

S− 1

2F fij

i∈I,j∈[m]

is

tight woven frame and for all f ∈ H, we have:

f =∑

i∈I,j∈[m]

⟨f, S

− 12

F fij

⟩S− 1

2F fij .

corollary 3.6. Let fiji∈I,j∈[m] be a woven frame for Hilbert space H and V,W are closed subspaces of Hsuch that V ∩W 6= φ and P denotes the orthogonal projection of H onto V ∩W . Then Pfiji∈I,j∈[m] is awoven frame for V ∩W .

In this section, we provide an example of woven frames in the Euclidean space R3, then from this wovenframe, we constitute a woven frame for smaller subspace of R3, by Theorem ?? and Corollary 3.6.

139

Z. Samadzadeh

example 3.7. Let ei3i=1 be the standard orthonormal basis for R3. Suppose there exist constants α > 0and β = 1√

1+α2. Also G,Q are the sets

G = gi6i=1 = βe1, αβe1, βe2, αβe2, βe3, αβe3, , Q = qi6i=1 = αβe1, βe1, αβe2, βe2, αβe3, βe3, ;

such that both of them are Parseval frames. Now, If we have:

C = minCσj s.t j ∈ [64]

, D = max

Dσj s.t j ∈ [64]

,

therefore G,Q are woven frames with universal bounds C and D. Now, let V1 = span e3i, e3i+1 and Pdenotes the orthogonal projection from R3 onto V1. Then by Theorem ??, Pgi6i=1 and Pqi6i=1 are wovenframes for V1.Also, suppose V2 = span e3i+1, e3i+2. Then V1 ∩ V2 = span e3i+1. Let P ′ be an orthonormal projectionof R3 onto span e3i+1. Thus by Corollary 3.6, P ′gi6i=1 and P ′qi6i=1 are woven frames for V1 ∩ V2 withsame bounds C and D.

References

[1] T. Bemrose, P. G. Casazza, K. Grochenig, M. C. Lammers, R. G. Lynch, Weaving Frames, J. Oper.Matrices. 2016, 10, 1093-1116.

[2] P. G. Casazza, G. Kutyniok, Frames of Subspaces, Contemp Math. Amer. Math. Soc. 2004, 345, 87-113.

[3] P. G. Casazza, G. Kutyniok, Sh. Li, Fusion frames and distributed processing, Appl. Comput. Harmon.Anal. 2008, 25, 114-132.

[4] P. G. Casazza, R. G. Lynch, Weaving properties of Hilbert space frames, J. Proc. SampTA. 2015, 110-114.

[5] O. Christensen, An introduction to frames and Riesz Basis, Birkhauser, Boston. 2016.

[6] I. Daubechies, A. Grossmann, Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 1986, 27,1271-1283.

[7] R. J. Duffin, A. C. Schaeffer, A class of nonharmonic Fourier series, J. Trans. Amer. Math. Soc. 1952,72, 341-366.

[8] A. Rahimi, Z. Samadzadeh, B. Daraby, Frame related operators for woven frames, International Journalof Wavelets and Multiresolution and Information processing (to appear).

[9] L. K. Vashisht and Deepshikha, Weaving K-frames in Hilbert spaces, Adv. Pure Appl. Math.arXiv:1710.09562v4.

[10] L. K. Vashisht, Deepshikha, S. Garg and G. Verma , On weaving fusion frames for Hilbert spaces, In:Proceedings of SampTA. 2017, 381-385.

140

Specification property for the iterated function systems

Z. Shabani*

(University of Sistan and Baluchestan, Zahedan, Iran)

Abstract: In this paper, we introduce the notion of specification property for the iterated functionsystems(IFSs) and prove that any expansive IFS(F) with shadowing and specification property has thetwo-sided limit shadowing property.

keywords. Iterated function systems, Specification property, Limit shadowing property

subject. 35B41, 37E30

1 Introduction

The specification property for a continuous map on a compact metric space X was introduced by Bowenin [3] to study the ergodic property of Axiom A diffeomorphisms. A mapping f : X → X has specificationproperty if one can approximate distinct pieces of orbits by single periodic orbit with a certain uniformity.It has been shown that if f has the specification property, then it is topologically mixing and the set of allperiodic points of f is dense in X. So, any mapping with the specification property is chaotic in the sense ofDevaney [1]. In this paper, we define the specification property for the iterated function systems and obtainsome results.

Suppose that (X, d) is a compact metric space and F = f1, f2, . . . , fm be a finite family of homeo-morphisms on X. We denote by F the semigroup generated by these maps. The dynamical system (F , X)is called an iterated function system(IFS). Here, we use the notation IFS(F) for the iterated function systemgenerated by F. The elements of F are called the generators of IFS(F).

Let Σm be space of two-sided infinite sequences of m symbols 1, . . . ,m i.e., Σm = 1, . . . ,mZ. Sup-pose that ω, ω′ ∈ Σm. Consider the standard metric ρ on Σm with ρ(ω, ω′) = 2−n, where n = min|k|;ωk 6=ω′k. For ω = (. . . , ω−1.ω0, ω1, . . .) ∈ Σm, the orbit of point x ∈ X under IFS(F) is a sequence fnω (x)n∈Z,where, f0

ω := id,fnω (x) := fωn−1 . . . fω0(x), f−nω (x) := f−1

ω−n . . . f−1ω−1

(x).

Let Am be the set of finite word of symbols 1, . . . ,m, i.e., if w ∈ Am, then w = w0 . . . wl−1, wherewi ∈ 1, . . . ,m for all i = 0, . . . , l − 1.

Bahabadi [2], introduced the shadowing property for iterated function systems. Let δ > 0 be given. Thesequence xii∈Z is called a (δ, ω)-pseudo orbit for IFS(F) for some sequence ω = (. . . , ω−1.ω0, ω1, . . .) ∈ Σm,if for any i ∈ Z,

d(fωi(xi), xi+1) < δ.

Clearly, for any x ∈ X the orbit fnω (x)n∈Z is a (δ, ω)-pseudo orbit. We say that a (δ, ω)-pseudo orbitxii∈Z is ε-shadowed by a point z ∈ X if there exists γ ∈ Σm such that for every i ∈ Z

d(f iγ(z), xi) < ε.

141

Z. Shabani

The IFS(F) is said to have shadowing property, provided that for every ε > 0, there exists δ > 0, such thatevery (δ, ω)-pseudo orbit can be ε-shadowed by some point z ∈ X.

definition 1.1. We say that an IFS(F) has specification property if for any ε > 0, there exists M(ε) > 0such that for any set x1, . . . , xk of points of X, any sequence a1 ≤ b1 < a2 ≤ b2 < . . . < ak ≤ bk of integerswith aj+1 − bj > M(ε), and any w(j) = waj . . . wbj−1 ∈ Am, (1 ≤ j ≤ k), there exists a point z ∈ X andω ∈ Σm with ωi = wi for any aj ≤ i ≤ bj − 1 such that

d(f iω(z), fi−ajw(j) (xj)) < ε,

for any aj ≤ i ≤ bj , 1 ≤ j ≤ k.

An IFS(F) is c-expansive if there is a constant c > 0(called an expansive constant) such that for everyarbitrary ω ∈ Σm, if the condition d(fnω (x), fnω (y)) < c holds for all n ∈ Z, then x = y.

A sequence xii∈Z is called a (negative) ω-limit pseudo orbit for IFS(F) for some ω ∈ Σm if

d(fωi(xi), xi+1)→ 0,

as (i → −∞) i → +∞. We say that an IFS(F) has the (negative) limit shadowing property if every(negative) ω-limit pseudo orbit for IFS(F) is (negative) limit shadowed by some point of z ∈ X, that is,there is sequence γ ∈ Σm such that

d(f iγ(z), xi)→ 0,

as (i→ −∞) i→ +∞.A sequence xii∈Z is called a ω-two-sided limit pseudo orbit for IFS(F) for some ω ∈ Σm if

d(fωi(xi), xi+1)→ 0,

as |i| → ∞. This pseudo orbit is said to be two-sided limit shadowed if there exist a point z ∈ X andγ ∈ Σm such that

d(f iγ(z), xi)→ 0,

as |i| → ∞. An IFS(F) has the limit shadowing property if every ω-two-sided limit pseudo orbit for IFS(F)is (strong) two-sided limit shadowed by some point of X.

2 Main results

In this section we state our main result.

lemma 2.1. Any expansive IFS with the shadowing property has both limit shadowing and negative limitshadowing property.

theorem 2.2. Any expansive IFS(F) with shadowing and specification property has the two-sided limitshadowing property.

example 2.3. Let X = a, b, c and a, b, c be three different points of X. With the discrete metric d, (X, d)is a compact metric space. Let f0, f1 be two cyclic permutations on X. that is,

f0(a) = b, f0(b) = c, f0(c) = a, f1(a) = c, f1(b) = a, f1(c) = b.

Then f0, f1 are homeomorphisms on X. Denote IFS(F) the iterated function system generated by f0, f1.It is easy to see that IFS(F) is c-expansive. Also IFS(F) has specification property and has shadowingproperty, so by Theorem 2.2 it has two-sided limit shadowing property.

142

Specification property for the iterated function systems

References

[1] N. Aoki, K. Hiraide, Topological Theory of Dynamical Systems, North-Holland Math. Library, North-Holland, Amsterdam, 1994.

[2] A.Z. Bahabadi, Shadowing and average shadowing properties for iterated function systems, GeorgianMathematical Journal, 22 (2015), pp. 179–184.

[3] R. Bowen, Equilibrium States and Ergodic Theory of Anosov Diffieomorphisms, Transactions of theAmerican Mathematical Society, 154 (1971), pp. 377–397.

143

O-cross Gram matrices with respect to g-frames

M. Shamsabadi*

(Hakim Sabzevari University, Sabzevar, Iran)

Abstract: The matrix representation of operators in Hilbert spaces is a useful tools in applications.It is important to present the matrix representation by using orthonormal bases, Riesz bases and frames.In this paper, we extend the matrix representation of operators by using g-frames and investigate theirinvertibility and stability.

keywords. U -cross Gram matrices; cross Gram matrices; g-frames; g-Riesz bases

subject. Primary 41A58; Secondary 43A35

1 Introduction

Frames, were introduced by Duffin and Schaeffer in 1952, are very important in signal processing, imageprocessing and many other fields. Recently, several generalization of frames, as continuous frames, fusionframes, g-frames [6] were presented. The concept of g-frames, which was introduced by Sun [6], is a gen-eralization of frames and continues by many authors [3, 2, 4, 7]. Throughout this paper, H is a Hilbertspace and Hii∈I a sequence of closed subspaces of H, where I is a subset of Z. Also, B(H,Hi) is thecollection of all bounded linear operators from H into Hi. Moreover, `2 is the set of scalars cii∈I suchthat

∑i∈I |ci|2 <∞.

A sequence Λ = Λi ∈ B(H,Hi) : i ∈ I is called a generalized frame or a g-frame for H with respectto Hii∈I if there exist constants A,B > 0 such that

A‖f‖2 ≤∑

i∈I‖Λif‖2 ≤ B‖f‖2, (f ∈ H). (51)

The numbers A and B are called the lower and upper g-frame bounds. It is called a g-Bessel sequence, ifthe right hand of (51) is satisfied. The family Λ is called [6]:

(i) a g-complete set for H with respect to Hii∈I if H = spanΛ∗i (Hi)i∈I .

(ii) a g-Riesz sequence for H with respect to Hii∈I if there are positive constants A and B such that

A∑

i∈I1‖gi‖2 ≤

∥∥∥∥∥∥∑

i∈I1Λ∗i gi

∥∥∥∥∥∥

2

≤ B∑

i∈I1‖gi‖2 ,

for all finite subset I1 of I and gii∈I1 ⊆ Hii∈I1 . Moreover, if it is a g-complete set, then it is calledg-Riesz basis.

144


(iii) a g-orthonormal system for H with respect to Hii∈I , if

⟨Λ∗i g,Λ

∗jg′⟩

= δij

⟨g, g

′⟩, (i, j ∈ I, g ∈ Hi, g

′ ∈ Hj).

(iiv) a g-orthonormal basis for H with respect to Hii∈I , if it is a g-orthonormal system for H with respectto Hii∈I and Λ∗i eiji∈I,j∈Ji is a basis for H, where eijj∈Ji is an orthonormal basis for Hi, for alli ∈ I.

It is easy to see that the space

(∑

i∈I⊕Hi

)

`2

=

fii∈I : fi ∈ Hi,

∑

i∈I‖fi‖2 <∞

with the inner product

〈fii∈I , gii∈I〉 =∑

i∈I〈fi, gi〉

is a Hilbert space. The synthesis operator of a g-Bessel sequence Λ = Λii∈I is defined by

TΛ :

(∑

i∈I⊕Hi

)

`2

→ H, TΛ (fii∈I) =∑

i∈IΛ∗i (fi).

The associated adjoint operator given by

T ∗Λ : H →(∑

i∈I⊕Hi

)

`2

, T ∗Λ(f) = Λifi∈I ,

is called the analysis operator. Also, the g-frame operator SΛ is defined by

SΛ : H → H, SΛf =∑

i∈IΛ∗iΛif.

For a g-frame Λ = Λii∈I , SΛ is a bounded, self adjoint and invertible operator and so we have the followingreconstruction formula

f =∑

i∈IΛ∗i Λif =

∑

i∈IΛ∗iΛif, (f ∈ H), (52)

where Λi = ΛiS−1Λ . See [6] for more details. The following theorem, which is used frequently, is easily

obtained.

theorem 1.1. Let Λ = Λii∈I be a g-Bessel sequences for H with respect to Hii∈I . Then

(1) Λ is a g-frame if and only if TΛ is onto.

(2) Λ is a g-Riesz basis if and only if TΛ is bijective.

definition 1.2. Let Λii∈I and Θii∈I be g-Bessel sequences for H with respect to Hii∈I . We callΘii∈I a dual g-frame of Λii∈I , if

f =∑

i∈IΛ∗iΘif, (f ∈ H).

In this case, Λii∈I and Θii∈I are also g-frames for H with respect to Hii∈I .

145

M. Shamsabadi

By (52), it is obvious that Λii∈I is a dual of Λii∈I which is called the canonical dual g-frame ofΛii∈I .

Let Λi ∈ B(H,Hi) and eijj∈Ji be orthonormal bases for Hi, where Ji is a subset of Z. Then thereare always elements

uij = Λ∗i eij , (i ∈ I, j ∈ Ji), (53)

such that

Λif =∑

j∈Ji〈f, uij〉 eij , (f ∈ H). (54)

The sequence uiji∈I,j∈Ji which is called the sequence induced by Λii∈I with respect to eiji∈I,j∈Jiplays a key role in this topic, see [6].

theorem 1.3. [6] Let Λii∈I ⊆ B(H,Hi) and uiji∈I,j∈Jibe defined by (53). Then

(1) Λii∈I is a g-frame (resp. g-Bessel sequence, g-Riesz basis, g-orthonormal basis) for H if and only ifuiji∈I,j∈Ji is a frame (resp. Bessel sequence, Riesz basis, orthonormal basis) for H.

(2) The g-frame operator of Λii∈I coincides with the frame operator of uiji∈I,j∈Ji .

(3) Λii∈I and Θii∈I are a pair of (canonical, weakly) dual g-frames if and only if their inducedsequences are a pair of (canonical, weakly) dual frames.

definition 1.4. Let Λ = Λi ∈ B(H,Hi) : i ∈ I and Θ = Θi ∈ B(H,Hi) : i ∈ I be g-Bessel sequences,and O ∈ B(H). The g-matrix representation GO,Λ,Θ, which the elements are in B

(∑i∈I⊕Hi, Hi

), given

by

(GO,Λ,Θ)ij = ΛiOΘ∗j

is called the O-cross Gram matrix. If O = IH, it is called cross Gram matrix and is denoted by GΛ,Θ. Weuse GΛ for GΛ,Λ; so called the Gram matrix. For More details see [5]

The standard matrix description of operators O using an ONB eii∈I is by constructing an matrix Mwith the entries Mjk = 〈Oek, ej〉. In [3] this concept was presented with Bessel sequences, Riesz bases andframes.

2 Main Results

In this section, we present some results about g-matrix representations.

lemma 2.1. Let Λ = Λii∈I and Θ = Θii∈I be two g-Bessel sequences in H with respect to Hii∈I .Also, let O ∈ B(H). The following assertions hold.

(1) GO,Λ,Θ = T ∗ΛOTΘ. In particular, the O-cross Gram matrix GO,Λ,Θ defines a bounded operator on(∑i∈I ⊕Hi

)`2

and ‖GO,Λ,Θ‖ ≤√BΛBΘ‖O‖.

(2) (GO,Λ,Θ)∗ = GO∗,Θ,Λ.

(3) If GO,Λ,Θ is a compact operator, then limi→∞ ‖ΛiOΘ∗i ‖ = 0.

(4) If∑

i∈I∑

j∈I

∥∥∥ΛiOΘ∗j

∥∥∥2<∞, then GO,Λ,Θ is compact.

proposition 2.2. Let Λ = Λii∈I be a g-Bessel sequence forH with respect to Hii∈I and u = uiji∈I,j∈Jiits induced Bessel sequence. There exists a unitary operator U ∈ B

((∑i∈I ⊕Hi

)`2, `2)

such that TΛ = TuU .

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theorem 2.3. Let Λ = Λii∈I and Θ = Θii∈I be g-Bessel sequences for H with respect to Hii∈I . Also,

let uΛ =uΛij

i∈I,j∈Ji

and uΘ =uΘij

i∈I,j∈Ji

be their induced Bessel sequences. Then for O ∈ B(H) the

following assertions hold.

(1) GO,Λ,Θ is invertible if and only if GO,uΛ,uΘis invertible.

(2) GO,Λ,Θ is the identity on(∑

i∈I ⊕Hi

)`2

if and only if GO,uΛ,uΘis the identity on `2.

(3) If GO,Λ,Θ is invertible, then Λ and Θ are g-Riesz sequences. Moreover, if O is an invertible operatorand Λ or Θ is g-frame, then

G−1O,Λ,Θ = G

O−1,Θ,Λ(55)

corollary 2.4. (1) Let Λ = Λii∈I and Θ = Θii∈I be g-frames. Then GO,Λ,Θ = I`2 if and only if Λand Θ are g-Riesz bases and Θi = ΛiOSΘ, for all i ∈ I.

(2) Let Λ = Λii∈I and Θ = Θii∈I be dual g-frames. If GO,Λ,Θ = I`2 , then Λ = Θ.

proposition 2.5. Let Λ = Λii∈I be a g-Bessel sequence in H with Bessel bound BΛ. If O,O1, O2 ∈ B(H)such that GO,Λ,Λ is an invertible operator, then the following assertions hold.

(1) If

‖O1 − IH‖ <1∥∥∥G−1

O,Λ,Λ

∥∥∥BΛ‖O‖, (56)

then GO,Λ,O1Λ and GO,O1Λ,Λ are also invertible.

(2) If

‖O∗1OO2 −O‖ <1∥∥∥G−1

O,Λ,Λ

∥∥∥BΛ

, (57)

then GO,O1Λ,O2Λ is also invertible. Moreover, if Λ is a g-frame, then

G−1O,O1Λ,O2Λ = T−1

Λ

∞∑

k=0

(O−1 (O∗1OO2 −O)

)kO−1T

Λ.

References

[1] A. Abdollahi, A. Rahimi, Some results on g-frames in Hilbert spaces, Turk. J. Math., 34, (2010), 695-704.

[2] A. A. Arefijamaal and S. Ghasemi, On characterization and stability of alternate dual of g-frames, Turk.J. Math., 37, (2013), 71-79.

[3] P. Balazs. Matrix-representation of operators using frames. Sampling Theory in Signal and ImageProcessing (STSIP), 7(1), (2008), 39-54.

[4] A. Najati, M. H. Faroughi, A. Rahimi, g-frames and stability of g-frames in Hilbert spaces, MethodsFunct. Anal. Topology., 4 (3), (2008), 271-286.

[5] M. Shamsabadi, A. Arefijamaal, O-cross Gram matrices with respect to g-frames, submitted.

[6] W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl., 322, (2006), 437-452.

[7] Y. C. Zhu, Characterizations of g-frames and g-bases in Hilbert spaces, Acta Math. Sin., 24, (2008),1727-1736.

147

On the Fourier algebra of a C∗-dynamical system

M. Shams Yousefi*

(University of Guilan, Rasht, Iran)

Abstract: Using the theory of C∗-dynamical systems C∗-valued Fourier Stieltjes algebras have beenstudied by the author. In this paper C∗-valued Fourier algebras will be defined as a closed ideal.

keywords. C∗-dynamical system, covariant representation, C∗-crossed product, Fourier- algebra, HilbertC∗-modules

subject. Primary 46L55; Secondary 46L05.

1 Introduction and Preliminaries

The Fourier and Fourier-Stieltjes algebras, A(G) and B(G), for a general locally compact group G, firststudied by P. Eymard in 1964, have played an important role in harmonic analysis and in the study of theoperator algebras generated by G [8]. He used the group representation and their corresponding C∗-algebras.The theory of positive definite functions is effective in his structure, which are introduced and studied muchearlier (see for instance [11]). It is well known that B(G) admits a natural Banach space structure for whichit is isometrically isomorphic to the dual space of the full group C∗-algebra C∗(G) associated with G. TheFourier algebra A(G) consists precisely of those functions on G which are the convolution product of twofunctions of L2(G) [8].

After that the Fourier-type algebras such as Herz algebras in 70’s was studied, that is the p–analogof the Fourier algebras, see [12], [13] and [9]. Also the p-analog of the Fourier–Stieltjes algebras by Rundein 2005 has been studied for 1 < p < ∞. He works on representations on QSLp-spaces and generalizedEymard Fourier-Stieltjes algebras [23]. Similar constructions on semigroups has been sorted by Amini andMedghalchi in [1] and the author in [24]. Some more generalization of these algebras to the other settings,for examples quantum groups [4] and [5], and groupoids have been studied, [16], [18]. In this paper wewould like to construct the vector valued Fourier–type algebras. There is also some efforts on vector valuedpositive definite functions in [14] and [22].

In [2], authors using a special type of the coefficients of equivariant representations of the a unitaldiscrete twisted C∗-dynamical system, defined a Fourier–Stieltjes Banach algebra. Their construction leadsto a Banach algebra which contains the classical Fourier–stieltjes algebra. Also they introduce a notion ofpositive definiteness and prove a Gelfand–Raikov type theorem.

In this paper, first we recall some information about C∗-dynamical systems and C∗-crossed products.Crossed products are built from locally compact group actions on C∗-algebras. Therefor the reader will needto have a bit of expertise in both of these subjects, we recommend [10] and [6]. The main idea of this paperis considering the C∗-crossed product as the generalization of group C∗-algebra and a generalized version ofits dual space as the C∗-valued Fourier–Stieltjes algebra, which mainly discussed in Section 2. Our approachto the dual space of the C∗-crossed product is different from [20], therefore we could make relation between

148


representations and the elements of this generalized dual space (Theorem 2.6), and so we could develop theclassic theory to the vector valued one.

Our construction is similar but different from [2]. The main idea of [2] to construct a vector valuedFourier-Stieltjes algebra is the fact that, any element of Fourier-Stieltjes algebra (respectively, positive coneof Fourier-Stieltjes algebra) induces a completely bounded (respectively, completely positive) map on fullgroup C∗-algebra and on the reduced group C∗- algebra, for example see [21]. But we focus on the duality ofthe full group C∗- algebra and the Fourier-Stieltjes algebra. Our construction gives us a Banach algebra ofvector valued functions on locally compact group. Moreover in [2], the C∗-dynamical systems is discrete andtwisted, but we follow [25] for the theory of C∗-dynamical systems. We will show that under some conditionsthere is a complete isomorphism (as Banach slgebras) between our construction and the construction studiedin [2] (See Theorem 2.14).

Also we will introduce a candidate for the vector valued Fourier algebras in Section 3, which has notstudied yet. Some classic results about Fourier and Fourier–Stieltjes algebra will be generalized in vectorvalued context. We will show that A(G,A) is a closed ideal of B(G,A) (Proposition 3.3).

All over this paper A is a unital C∗-algebra and G is a locally compact group, further assumption willbe mentioned.

1.1 Hilbert C∗-modules

Let M be a module over the C∗algebra A, a right action of an element a ∈ A on M is denoted by x · a,where x ∈M .

definition 1.1. A pre-Hilbert A-module is a (right) A-module Mequipped with a sesquilinear form 〈·, ·〉 :M ×M → A, with the following properties :

(i) 〈x, x〉 ≥ 0 for any x ∈M ;

(ii) 〈x, x〉 = 0implies that x = 0;

(iii) 〈x, y〉 = 〈y, x〉∗for any x, y ∈M ;

(iv)〈x, y · a〉 = 〈x, y〉 · a.The map 〈·, ·〉is called an A-valued inner product .

By taking a quotient of Mon the kernel of 〈·, ·〉, and considering ‖x‖ := ‖〈x, x〉 12 ‖ as a norm on the

quotient space, we will have a normed pre–Hilbert A-module.

definition 1.2. A pre-Hilbert A-module M , is called a Hilbert C∗-module, if it is complete with respect tothe norm defined above .

If A is a C∗-algebra, then it is itself a Hilbert C∗-module with the natural following definition:

〈x, y〉 := x∗y, x, y ∈ A.

Also each Hilbert space is a Hilbert C-module.

For more on Hilbert C-modules see [15].

In the following, we study the generalized version of Gelfand–Naimark-Segal construction for HilbertC∗-modules.

Let A and B be C∗-algebras and φ : A→ B be a linear map. We say that φ is completely positive if

∑

i,j

b∗iφ(a∗i aj)bj ≥ 0,

in B, for each n ≥ 1, and each a1, · · · , an ∈ A and b1, · · · , bn ∈ B. This is known to be equivalent toφ being completely positive in the sense of operator spaces, that is to the positivity of all amplificationsφn : Mn(A)→Mn(B) [17, page 463].

The following lemma, that will be used a little further , follows from [17, Theorem 5.2].

149

M. Shams Yousefi

lemma 1.3 (GNS-construction). Let A and B be a unital C∗-algebras, let φ : A −→ B be a completelypositive linear map. Then there is a Hilbert B-module Hφ and vector ξ ∈ H, and a representation πφ : A→B(Hφ), such that φ(a) = 〈πφ(a)ξ, ξ〉, and the set of all elements of the from πφ(a)ξ · b, for a ∈ A and b ∈ B,span a dense subspace of H.

We need the following notions of tensor products of Hilbert C∗-modules. There are two useful waysto construct tensor products of Hilbert C∗-modules. The first usually calls exterior tensor product , whichhas the structure that one expect from algebraic tensor product, and the second one that calls internal (orinterior) tensor product , looks less natural but more useful in applications. In the following we describethe second one, the interested reader can find more about this topic in [15].

Suppose X and Y are Hilbert A-modules. We define L(X,Y ) to be the set of all maps T : X → Y , forwhich there exists T ∗ : Y → X, such that

〈Tx, y〉 = 〈x, T ∗y〉, x ∈ E, y ∈ F,

it is easy to see that T must be A-linear and bounded. We call the set L(X,Y ), the set of all adjointablemaps from X to Y , when X = Y we denote L(X,Y ) by L(X).

By UL(X) we mean all unitary adjointable maps.

definition 1.4. Let A and B be two M1 and M2 Hilbert C∗-modules over A and B, respectively, andρ : A −→ L(M2) a ∗-homomorphism. Consider the B-valued inner product

〈e1 ⊗ e2, f1 ⊗ f2〉 = 〈e2 ⊗ e2, ρ(〈e1, f1〉)f2〉, ei, fi ∈Mi. (58)

Let N be the submodule of isotropic vectors in the algebraic tensor product (M1 ⊗M2). Then thecompletion of the quotient space (M1 ⊗M2)/N with respect to the product 58 equipped with the obviousHilbert C∗-module structure is called the interior tensor product M1 ⊗ρM2. The action of B is defined bythe formula (e1 ⊗ e2) · b = e1 ⊗ e2 · b also, e1 ⊗ ρ(a)e2 = e1 · a⊗ e2.

1.2 C∗-dynamical systems

A C∗-dynamical system achieved by group acting (by automorphisms) on a C∗-algebra. Out of a dynamicsystem we can built a C∗-algebra, which will be discussed in this section. We follow [25] in definitions andnotations.

definition 1.5. A C∗-dynamical system is a triple (A,G, α) consisting of a C∗-algebra A, a locally compactgroup G and a continuous homomorphism α : G→ AutA, where AutA denotes all ∗–isomorphisms on A.

example 1.6. Groups and C∗-algebras are by themselves degenerate examples of dynamical systems. Ev-ery locally compact group G gives rise to a dynamical system (C, G, id), similarly every C∗-algebra A isassociated to a dynamical system with G trivial, (A, e, id).

definition 1.7. Let (A,G, α) be a C∗-dynamical system. Then a covariant representation of (A,G, α) is apair (π, U), consisting of a representation π : A→ B(H) and a unitary representation U : G→ U(H) on thesame Hilbert space H, such that

π(αs(a)) = Usπ(a)U∗s .

In the following we give the reasonable way to represent a dynamical system on a Hilbert modules.

definition 1.8. Let (A,G, α)be a C∗-dynamical system. Then a covariant homomorphism of (A,G, α)is a pair (π, U), consisting of a homomorphism π : A → L(X) and a strongly continuous unitary-valuedhomomorphism U : G→ UL(X) on the same Hilbert module X, such that

π(αs(a)) = Usπ(a)U∗s .

It is clear that covariant homomorphisms are the generalization of covariant representations.

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example 1.9. Let Gact on itself by left translation, and lt : G → AutC0(G), be the associated dynamicalsystem action. Let M : C0(G)→ B(L2(G)) be given by pointwise multiplication :

M(f)h(s) := f(s)h(s),

and λ : G → U(L2(G)) be the left regular representation. Then (M,λ) is a covariant representation of(C0(G), G, lt), where the Hilbert space L2(G) has been considered as a C-module in the natural way.

example 1.10. Let ρ : A→ B(Hρ) be any representation of A on a Hilbert space Hρ. Then define IndGe ρto be the pair (ρ, U) of representation on the Hilbert space L2(G,Hρ), where

ρ(a)h(r) := ρ(α−1r (a))(h(r))

andUsh(r) := h(s−1r)

then by [25, 2.14], IndGe ρ := (ρ, U) is a covariant representation .

Now we define the vector valued group C∗-algebra. Group C∗-algebras and crossed products are sortof completion of families of continuous functions. For readers interested in vector valued integration wesuggest appendix B in [25].

If f, g ∈ Cc(G,A), then

f ∗ g(s) :=

∫f(r)αr(g(r−1s))dµ(r)

defines an element of Cc(G,A), and called the convolution of f and g. And it is not hard to check thatf ∗ (g ∗ h) = (f ∗ g) ∗ h. Also a similar computation shows that

f∗(s) := ∆(s−1)αs(f(s−1)∗)

is a involution on Cc(G,A) to make Cc(G,A) a ∗-algebra. Furthermore

‖f‖1 :=

∫‖f(s)‖dµ(s)

is a norm on Cc(G,A) and‖f‖1 = ‖f∗‖1

and ‖f ∗ g‖1 ≤ ‖f‖1‖g‖1.

definition 1.11. A ∗-homomorphism π : Cc(G,A) → B(H), for some Hilbert space H, is called a ∗-representation of Cc(G,A) on H.

proposition 1.12. [25, Proposition 2.23] Suppose that (π, U) is a covariant representation of (A,G, α)onH. Then

π o U(f) :=

∫π(f(s))Usdµ(s)

defines a L1–norm decreasing ∗-representation of Cc(G,A)on H, called the integrated form of (π, U)

In this case we have

〈π o U(f)ξ, η〉 = 〈∫π(f(x))U(x)dxξ, η〉

=

∫〈π(f(x))U(x)ξ, η〉dx.

The definition of integrated form of a covariant homomorphism is similar.

151

M. Shams Yousefi

proposition 1.13. [25, Lemma 2.27] Suppose that (A,G, α) is a dynamical system and for each f ∈ Cc(G,A)we define

‖f‖ := sup‖π o U(f)‖ : (π, U) is a covariant rep. of (A,G, α).Then ‖ · ‖is a norm on Cc(G,A)called the universal norm, which is dominated by ‖ · ‖1-norm, and thecompletion of Cc(G,A) with respect to ‖ · ‖ is a C∗-algebra called the crossed product of A by G, denotedby Aoα G.

In the following we give the universal form of the non degenerate representation of Aoα G.

proposition 1.14. [25, Proposition 2.39] Suppose that (A,G, α) is a C∗-dynamical system and that X is aHilbert B-module. If (π, U) is a covariant homomorphism of (A,G, α) into L(X), then the integrated form

π o U(f) :=

∫π(f(s))Usdµ(s)

is a well-defined operator in L(X), and π o U extends to a homomorphism of A oα G into L(X) which isnon degenerate whenever π is non degenerate.

proposition 1.15. [25, Proposition 2.40]The map sending a covariant pair (π, U) to its integrated formπ o U is a one to one correspondence between non degenerate covariant homomorphism of (A,G, α) andnon degenerate representations of Aoα G, on Hilbert modules. This correspondence preserves direct sums,irreducibility and equivalence.

For covariant representations (π1, U1) and (π2, U2) of the C∗-dynamical system (A,G, α), on HilbertA-modules X1 and X2, by the tensor product (π1, U1) ⊗ (π2, U2), we mean a covariant representation of(A,G, α) on the internal tensor product Hilbert A-module X1 ⊗π2 X2, as follows.

For a ∈ A, we let (π1 ⊗ π2)(a) ∈ L(X1 ⊗π2 X2) be determined by

[(π1 ⊗ π2)(a)](x1 ⊗ x2) = ρ1(a)x1⊗ x2, forx1 ∈ X1, andx2 ∈ X2.

It is easily checked that the associated map π1 ⊗ π2 : A −→ L(X1 ⊗π2 X2) is a representation of A onX1 ⊗π2 X2.

Also, for each g ∈ G, it is straightforward to check that for the map defined by

[(U1 ⊗ U2)(g)](x1 ⊗ x2) = U1(g)x1 ⊗ U2(g)x2, forx1 ∈ X1 andx2 ∈ X2,

we have (U1 ⊗ U2)(g) is in UL(X1 ⊗π2 X2) and we have the following lemma,

lemma 1.16. Let (π, U)and (π′, U ′)be covariant representation of (A,G, α), then so is (π ⊗ π′, U ⊗ U ′).Proof. This is a special case of [2, Proposition 2.1].

2 C∗-valued Fourier–Stieltjes algebras

In this section, we will define the C∗-valued Fourier Stieltjes algebras. In this way we use the structure ofAoα G as the generalized version of group C∗-algebra C∗(G).

Since the crossed product AoαG does not contain a copy of either A or G, working with the multiplieralgebra M(Aoα G) will be useful .

Now let π o U be the extension of π o U to M(A oα G). Then for each ξ, η in Hilbert A-module H,and s ∈ G, we define the coefficient function of the representation π o U , as

f(s) = 〈π o U(iG(s))ξ, η〉,where iG is the natural embedding of G in M(Aoα G).

When (π, U) is non degenerate, then π o U(iG(s)) = U(s), so the coefficient function has a simplerform as f(s) = 〈U(s)ξ, η〉. We will show that f is continuous bounded function from G to A, namely itbelongs to Cb(G,A) (Proposition 2.2).

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definition 2.1. By the C∗-valued Fourier–Stieltjes algebra, which is denoted by B(G,A), we mean the setof all functions of the form f(x) = 〈U(x)ξ, η〉, where U is a (class of) unitary group representation on U(H),for some Hilbert A-module H, which is covariant to a non degenerate representation π : A −→ B(H).

With the same notation as above, all functions of the form f(x) = 〈U(x)ξ, ξ〉, is denoted by P (G,A).

theorem 2.2. Let (A,G, α) be a C∗-dynamical system, then B(G,A) is a linear subspace of Cb(G,A),which is a linear span of the positive cone P (G,A).

Now we put the functional norm on B(G,A) as follows. Let f = 〈U(·)ξ, η〉 belongs to B(G,A), thenwe define :

‖f‖ = sup

∥∥∥∥∫〈π(g)U(x)ξ, η〉dx

∥∥∥∥ ,

where supremum is taken over all g ∈ Cc(G,A), with norm defined in Proposition 1.13 less than or equal to1. In this case, by the definition, for each f ∈ B(G,A)we have

‖f‖ = sup

∥∥∥∥∫〈π(g)U(x)ξ, η〉dx

∥∥∥∥= sup ‖〈π o U(g)ξ, η〉‖≤‖ξ‖‖η‖

For a function u : G → A and a ∈ A, we define a · u(x) = au(x), u∗(x) = u(x)∗, u(x) = u(x−1)∗ andu(x) = u(x−1).

proposition 2.3. Let (A,G, α) be a C∗-dynamical system, and let u ∈ B(G,A) and a ∈ A. Then a · u, u∗,u, and u belongs to B(G,A).

proposition 2.4. Let (A,G, α) be a C∗-dynamical system, then B(G,A) has a copy contained in B(AoαG,A), a generalized notion of the dual space of Aoα G.

example 2.5. It is easy to see that, for the degenerate dynamical system (C, G, id), the crossed productAoαG is the group C∗-algebra, and the B(G,A)defined above is nothing except the Fourier–Stieltjes algebraB(G).

In the following we give some more explicit characterization for Proposition 2.4.

theorem 2.6. Let (A,G, α) be a C∗-dynamical system, then CP(Aoα G,A) = P (G,A).

definition 2.7. We say that the C∗-algebra A, is decomposable, if each completely bounded map φ :Aoα G→ A, can be written as φ1 − φ2 + i(φ3 − φ4), where for i = 1, · · · , 4, φi is completely bounded andcompletely positive maps.

example 2.8. Injective von Neumann algebras are decomposable [19, Corollary 2.6].

corollary 2.9. Let (A,G, α) be a C∗-dynamical system, and Let A be a decomposable C∗-algebra thenCB(Aoα G,A) ∼= B(G,A).

theorem 2.10. Let (A,G, α) be a C∗-dynamical system, then B(G,A) is a Banach algebra.

remark 2.11. By the definition of the multiplication of two elements in B(G,A) and notations as above,if π′ comes from the natural module action of A on X ′, we have f g = fg.

theorem 2.12. Let (A,G, α) be a C∗-dynamical system. Then

1. B(G) embeds in B(G,A) continuously.

2. B(G,A) is unital.

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2.1 Comparing with the construction of Erik Bedos and Roberto Conti

In [2], the authors associate a Banach algebra of (special type) coefficients of equivariant representationsof the system, to a unital discrete twisted C∗-dynamical system. They show that this Fourier–Stieltjesalgebra is a generalization of the Eymard’s classical Fourier–Stieltjes algebra, and embeds continuously inthe Banach algebra of completely bounded multipliers of the (reduced or full) C∗-crossed product of thesystem. Which is a known result in clssic Harmonic analysis for B(G), for example see [21]. This approachis the main aim of the authors to construct the generalized Fourier–Stieltjes algebra.

They introduce a notion of positive definiteness and prove a Gelfand–Raikov type theorem Also theypropose a definition of amenability for C∗-dynamical systems

In this section we would like to compare our construction of B(G,A) and the construction of Bedos-Conti.

The quadruple Σ = (A,G, α, σ) is denoted a twisted unital discrete C∗-dynamical system, where A is aunital C∗-algebra, G is a discrete group with identity e and (α, σ) is a twisted action of G on A, that is, αis a map from G into Aut(A), as our previous notion in Section 1.2, and σ is a map from G×G into U(A),satisfying

αgαh = Ad(σ(g, h))αgh

σ(g, h)σ(gh, k) = αg(σ(h, k))σ(g, hk)

σ(g, e) = σ(e, g) = 1

for all g, h, k ∈ G.

If σ is trivial, then Σ is an ordinary C∗-dynamical system.

definition 2.13. By an equivariant representation of Σ = (A,G, α, σ) on a Hilbert A-module X we willmean a pair (ρ, v) where ρ : A −→ L(X) is a representation of A on X and v is a map from G into thegroup UL(X) consisting of all C-linear, invertible, bounded maps from X into itself, which satisfy:

1. ρ(αg(a)) = v(g)ρ(a)v(g)−1 , g ∈ G, a ∈ A,

2. v(g)v(h) = adρ(σ(g, h))v(gh) , g, h ∈ G,

3. v(g)v(h) = adρ((g, h))v(gh) , g, h ∈ G,

4. v(g)(xa) = (v(g)x)αg(a) , g ∈ G, x ∈ X, a ∈ A.

When σ is trivial, then v is covariant with ρ, see [3]. For more about twisted systems and theirassociated C∗- crossed products the reader could see [3].

Let (ρ, v) be an equivariant representation of Σ on a Hilbert A-module X and let x, y ∈ X. Then defineTρ,v,x,y : G×A→ A by

Tρ,v,x,y(g, a) = 〈x, ρ(a)v(g)y〉, for a ∈ A, g ∈ G.

The Bedos-Conti Fourier–Stieltjes algebra, B(Σ), denotes the set of all maps from G×A into A of theform Tρ,v,x,y for some equivariant representation (ρ, v) of Σ on a Hilbert A-module X and x, y ∈ X.

They define a norm on B(Σ) by letting ‖T‖ denote the infimum of the set of values ‖x‖‖y‖ associatedwith the possible decompositions of T of the form T = Tρ,v,x,y.

For more details about the multiplication of B(Σ) see [2, Section 3]. With these structure of norm andmultiplication, the Bedos-Conti Fourier–Stieltjes algebra B(Σ), is a unital Banach algebra, [2, Proposition3.1]. In the following theorem, we show that for ordinary twisted unital discrete C∗-dynamical system, B(Σ)and B(G,A) are the same (as Banach algebras).

theorem 2.14. Let Σ = (A,G, α, σ) be a ordinary twisted unital discrete C∗-dynamical system. Then B(Σ)is isomorphic to B(G,A) as Banach algebras. If A is a decomposable C∗-algebra, then B(Σ) is completelyisomorphic to B(G,A).

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3 C∗-valued Fourier algebras

Fourier–algebra occurs naturally in the harmonic analysis of locally compact groups. It plays an importantrole in the duality theories of these groups. Fourier algebra is a generalization of the algebra of all Fouriertransforms of absolutely integrable functions on a locally compact abelian group.

In this section we define the C∗-valued Fourier algebra, A(G,A), which is the generalized version ofEymard’s Fourier–algebra.

Let (A,G, α) be a C∗- dynamical system, then it is not hard to see that with the natural actions,

L2(G,A)×A −→ L2(G,A), f, a→ r → a∗h(r),

and

L2(G,A)× L2(G,A) −→ L2(G,A), f, g →∫f(s)g(s)∗ds,

L2(G,A) is a Hilbert A -module.We put ρ : A −→ B(A), a→ La, and

ρ : A −→ B(L2(G,A)), a→ h→ r → ρ(αr−1(a))(h(r)),

and,U : G −→ B(L2(G,A)), s→ h→ r → αs(h(s−1r)).

Then U is a unitary representation and it is covariant to ρ. On the other hand for each f, ξ ∈ Cc(G,A) andr ∈ G , we have

(ρo U)(f)(ξ)(r) =

∫(ρ(f(s))Us)(ξ)(r)ds

=

∫ρ(f(s))(Us(ξ))(r)ds

=

∫ρ(αr−1(f(s)))αs(ξ(s

−1r))ds

=

∫αr−1(f(s))αs(ξ(s

−1r))ds

= (αr−1 f) ∗ ξ(r),

also for each s ∈ G and η in Cc(G,A) we have η(s)∗ = αs(η∗(s−1))∆(s−1).

〈U(s−1)ξ, η)〉 =

∫U(s−1)ξ(r)η(r)∗dr

=

∫αs−1ξ(sr)η(r)∗dr

=

∫αs−1ξ(sr)αr(η

∗(r−1)∆(r−1))dr

=

∫αs−1ξ(sr)αr(η(r−1))dr

Also we have

ξ ∗ η(s) =

∫ξ(r)αr(η(r−1s))dr

=

∫ξ(sr)αsr(η(r−1))dr

= αs

∫αs−1ξ(sr)αr(η(r−1))dr

= αs(〈U(s−1)ξ, η〉)

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M. Shams Yousefi

so there is a version of classical relation between convolution and representation in C∗-valued case:

〈U(s−1)ξ, η〉 = αs−1(ξ ∗ η(s)) (59)

definition 3.1. LetA be a C∗-algebra, and let (A,G, α) be a C∗-dynamical system, then by the vector valuedfourier algebra we mean the set of all function of the form f(x) = 〈U(s−1)ξ, η)〉, for some ξ, η ∈ L2(G,A),and U defined above, with the norm and multiplication comes from B(G,A).

remark 3.2. When (C, G, id) is the degenerate C∗-dynamical system our construction is the known EymardFourier algebra.

proposition 3.3. Let (A,G, α) be a C∗-dynamical system, then A(G,A) is a closed ideal of B(G,A).

References

[1] M. Amini and A. R. Medghalchi, Restricted algebras on inverse semigroups. I. Representation theory,Math. Nachr., vol. 279, no. 16, (2006), 17391748.

[2] E. Bedos, R. Conti, The Fourier-Stieltjes algebra of a C -dynamical system, Int. J. Math. 27, no. 5(2016) 1650050(50 pages).

[3] E. Bedos and R. Conti, On discrete twisted C∗-dynamical systems, Hilbert C∗-modules and regularity,Munster J. Math. 5 (2012) 183208.

[4] J. de Canniere, M. Enock and J.-M. Schwartz, Algebres de Fourier associes a une alg‘ebre de Kac,Math. Ann. 245 (1979) 122.

[5] J. de Canniere, M. Enock and J.-M. Schwartz, Sur deux resultats danalyse harmonique non-commutative: Une application de la theorie des algebres de Kac, J. Operator Th. 5 (1981) 171194.

[6] J. B. Conway, A Course in Operator Theory, American Mathematical Society, 2000.

[7] J. J. Diestel, Jr. Uhl, Vector measures, Amer. Math. Soc. Mathematical Surveys and MonographsVolume: 15; 1977.

[8] P. Eymard , L,algebra de Fourier d,un groupe localment compact, Bull. Soc. Math. France, 92,(1964),181–236.

[9] A. Figa-Talamanca, Translation invariant operators in Lp, Duke Math. J. 32 (1965), 495-501.

[10] J. B. Folland, A course in Abstract Harmonic Analysis, CRC Press, 1995.

[11] R. Godement, Les fonctions de type positive et la theorie des groupes, Trans. Amer. Math. Soc. 63(1948), 1-84.

[12] C. Herz, The theory of p-spaces with an application to convolution operators, Trans. Amer. Math. Soc.154 (1971), 69–82.

[13] C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), 91–123.

[14] H. Hongyu, On matrix valued square integrable positive definite functions, Monatsh. Math. 177(2015), 437449.

[15] C. Lance, Hilbert C∗-modules, London Mathematical Society, 1995.

[16] K. J. Oty, FourierStieltjes algebras of r-discrete groupoids, J. Operator Theory, 41 (1999) 175197.

[17] W. Paschke, Inner product modules over B∗-algebras, Tran. Amer. Math. Soc,182(1973), 443–468.

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[18] A. L. T. Paterson, The Fourier algebra for locally compact groupoids, Canad. J. Math. 56 (2004)12591289.

[19] V. Paulsen, Completely bounded maps on C∗-algebras and invariant operator rings, Proc. Amer. Math.Soc., 86(1) (1982), 91–96.

[20] G. K. Pedersen, C∗-algebras and their auotomorphism groups, London Mathematical Society, 1978.

[21] G. Pisier, Introduction to Operator Space Theory, London Mathematical Society Lecture Notes Series,Vol. 294 (Cambridge University Press, Cambridge, 2003).

[22] P. Ressel and W. J. Ricker, Vector-valued positive definite function, the Berg-Maserick theorem, andapplications, math. scand. 90 (2002), 289319.

[23] V. Runde, Representations of locally compact groups on QSLp-spaces and a p-analog of the Fourier-Stieltjes algebra, Pacific J. Math. 221 (2005), 379–397.

[24] M. Shams Yousefi, p-analog of the semigroup Fourier Stieltjes algebras, Iranian J. math. Sci. and Inf,10(2), (2015), 55-66.

[25] D. P. Williams, Crossed Products of C∗-algebras, American Mathematical Society, Vol 134, 2007.

157

Module Johnson amenability of Banach algebras

S. F. Shariati*, A. Pourabbas (Amirkabir University of Technology, Tehran, Iran)A. Sahmi (Ilam University, Ilam, Iran)

Abstract: In this paper we introduce the new notion module Johnson amenability for a Banach alge-bra which is a Banach module over another Banach algebra with compatible actions. We study relationsbetween this new notion to module pseudo amenability, module approximate amenability and also Johnsonpseudo contractibility. We characterize the module Johnson amenability of `1(S) as an `1(E)-module, for aninverse semigroup S with subsemigroup E of idempotents. We investigate the module Johnson amenabilityof `1(S), whenever S is Brandt semigroup or bicyclic semigroup or N with maximum as its product. Weshow that for every non-empty set I, MI(C) as an A-module under this new notion is forced to have a finite

index, where A =

[ai,j ] ∈MI(C) | ∀i 6= j, ai,j = 0

.

keywords. module Johnson amenability, Johnson pseudo-contractibility, module amenability

subject. 46H20, 46H25, 43A10

1 Introduction

The concept of module amenability for a class of Banach algebras introduced by Amini [1]. He showed thatfor an inverse semigroup S, `1(S) is an `1(E)-module amenable if and only if S is amenable, where E is theset of idempotents [1, Theorem 3.1]. Some modificated notion of module amenability like module pseudoamenability, module pseudo-contractibility and module approximately amenability have been introduced,see [5], [8]. Bodaghi et al. showed that for an inverse semigroup S, `1(S) as an `1(E)-module is modulepseudo-amenable if and only if S is amenable [5, Theorem 3.13(i)]. Also the same result holds for the moduleapproximately amenability [8, Theorem 3.9].

The notion of Johnson pseudo-contractibility for a Banach algebra was introduced by second and thirdauthors, which is a weaker notion than amenability and pseudo-contractibility but it is stronger than pseudo-amenability [11]. A Banach algebra A is called Johnson pseudo-contractible, if there exists a not necessarilybounded net (mα) in (A⊗A)∗∗ such that a ·mα = mα · a and π∗∗A (mα)a → a for every a ∈ A. They alsoshowed that for a locally compact group G, M(G) is Johnson pseudo-contractible if and only if G is discreteand amenable [11, Proposition 3.3]. They characterized the Johnson pseudo-contractibility of `1(S), where Sis a uniformly locally finite inverse semigroup [10, Theorem 2.3]. They showed that for a Brandt semigroupS = M0(G, I) over a non-empty set I, `1(S) is Johnson pseudo-contractible if and only if G is amenableand I is finite [10, Theorem 2.4].By consideration these notions, we generalize the concept of Johnson pseudo-contractibility for a class ofBanach algebras that are modules over another Banach algebra with compatible actions.

In part two of this paper, we define the module Johnson amenability for a Banach algebra A which isa Banach A-module. First we show that the module Johnson amenability is a stronger notion than module

158


pseudo-amenability and module approximately amenability and it is a weaker notion than module pseudo-contractibility. Next for an inverse semigroup S with subsemigroup E of idempotents we characterize themodule Johnson amenability of `1(S) as an `1(E)-module with amenability of S.

In part three, we provide some examples to distinguish our new notion with the Johnson pseudo-contractibility. Finally as an application we show that for every non-empty set I, the Banach algebra ofI × I-matrices over C, MI(C) as an A-module is module Johnson amenable if and only if I is finite, where

A =

[ai,j ] ∈MI(C) | ∀i 6= j, ai,j = 0

.

2 Module Johnson amenability

Let A and A be Banach algebras such that A is a Banach A-bimodule with the following compatible actions:

α · (ab) = (α · a)b, (ab) · α = a(b · α) (a, b ∈ A, α ∈ A).

Let X be a Banach A-bimodule and a Banach A-bimodule with the compatible actions:

α · (a · x) = (α · a) · x, a · (α · x) = (a · α) · x, (a · x) · α = a · (x · α),

for every a ∈ A, α ∈ A and x ∈ X and similarity for the right or two side actions. Then we say that X isa Banach A-A-module. If moreover α · x = x · α for every a ∈ A, x ∈ X, then X is called a commutativeBanach A-A-module. If X is a commutative Banach A-A-module, then so is X∗, where the actions of Aand A on X∗ are defined as follows:

〈α · f, x〉 = 〈f, x · α〉, 〈a · f, x〉 = 〈f, x · a〉 (α ∈ A, a ∈ A, x ∈ X, f ∈ X∗),and similarity for the right actions. Let A and A be as above and X be a Banach A-A-module. A boundedmap D : A → X is called a module derivation if

D(a± b) = D(a)±D(b), D(ab) = D(a) · b+ a ·D(b) (a, b ∈ A),

andD(α · a) = α ·D(a), D(a · α) = D(a) · α (a ∈ A, α ∈ A).

When X is commutative, every x ∈ X defines a module derivation

Dx(a) = a · x− x · a (a ∈ A),

These are called inner module derivations. A is called module amenable as an A-module, if for any commu-tative Banach A-A-module X, every module derivation D : A → X∗ is inner [1, Definition 2.1].

Let A be a Banach A-module and let A⊗AA be the projective module tensor product of A and A,

which is isomorphic to the quotient spaceA⊗AIA

, where IA is the closed linear span of

a · α⊗ b− a⊗ α · b | α ∈ A, a, b ∈ A

in A⊗A. Also consider the closed ideal JA of A generated by

(a · α)b− a(α · b) | α ∈ A, a, b ∈ A.

We denote IA and JA by I and J repectively, unless otherwise specified. So I is a A-submodule and aA-submodule of A⊗A, J is a A-submodule and a A-submodule of A, and both of the quotients A⊗AA andAJ are A-module and A-module. Consider the product map ωA : A⊗A → A defined by a⊗ b 7→ ab for every

a, b ∈ A and let ωA : A⊗AA →AJ be its induced product map defined by ωA(a ⊗ b + I) = ab + J . It is

clear that ωA and ω∗∗A :(A⊗A)∗∗

I⊥⊥A→ A∗∗J ⊥⊥ are both A-module morphisms and A-module morphisms.

Now we introduce the new notion of this work:

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S. F. Shariati*, A. Pourabbas and A. Sahmi

definition 2.1. Let A be a Banach A-module. Then A is called module Johnson amenable, if there existsa not necessarily bounded net (mα) in (A⊗AA)∗∗ such that for every a ∈ A

(i) a · mα = mα · a,

(ii) ω∗∗A (mα) · a+ J → a+ J ⊥⊥ in A∗∗J⊥⊥ .

lemma 2.2. Let A be a Banach A-module. If A is Johnson pseudo-contractible, then A is module Johnsonamenable.

An element M ∈ (A⊗AA)∗∗ is called a module virtual diagonal if ω∗∗A (M)·a = a+J ⊥⊥ and M ·a = a·Mfor every a ∈ A [9].

lemma 2.3. Let A be a Banach A-module. If A has a module virtual diagonal, then A is module Johnsonamenable.

Let A be a Banach A-module. A Banach algebra A is said to be module pseudo-amenable (modulepseudo-contractible), if there exists a net (uj) in A⊗AA such that ωA(uj)·a+J → a+J and uj ·a−a·uj → 0(uj · a− a · uj = 0) for every a ∈ A [5, Definition 2.1], [5, Definition 2.2].

proposition 2.4. Let A be a Banach A-module. If A is module Johnson amenable, then A is modulepseudo amenable.

Let A be a Banach algebra and an A-bimodule with compatible actions. Then A is module approxi-mately amenable (as an A-module) if for every commutative BanachA-A-module X, every module derivationD : A → X∗ is approximately inner [8, Definition2.1].

corollary 2.5. Let A be a Banach A-module with bounded approximate identity. If A is module Johnsonamenable, then A is module approximately amenable.

proposition 2.6. Let A be a Banach A-module. If A is module pseudo-contractible, then A is moduleJohnson amenable.

Following [4, §2], let A be a Banach A-module with compatible action and let ϕ ∈ ∆A ∪0, where ∆A

is a character space of A. Consider a linear map φ : A → A such that for every a ∈ A and α ∈ A

φ(ab) = φ(a)φ(b), φ(a · α) = φ(α · a) = ϕ(α)φ(a).

A bounded linear functional m : A∗ → C is called a module (φ, ϕ)-mean on A∗ if m(f · a) = ϕ φ(a)m(f),m(f · α) = ϕ(α)m(f) and m(ϕ φ) = 1 for every f ∈ A∗, a ∈ A and α ∈ A. We say A is module(φ, ϕ)-amenable if there exists a module (φ, ϕ)-mean on A∗.proposition 2.7. Let A be a Banach A-module such that a · α = ϕ(α)a for every a ∈ A and α ∈ A, whereϕ, φ be as above. If A is module Johnson amenable moreover α · mα = mα · α for every α ∈ A (mα as inDefinition 2.1), then A is module (φ, ϕ)-amenable.

proposition 2.8. Let A be a Banach A-module. If A∗∗ is module Johnson amenable, then A is moduleJohnson amenable.

proposition 2.9. Let A and B be Banach A-modules. Suppose that Ψ : A → B is a continuous epimorphismsuch that Ψ(α · x) = α · Ψ(x) and Ψ(x · α) = Ψ(x) · α for every α ∈ A and x ∈ A. If A is module Johnsonamenable, then B is module Johnson amenable.

corollary 2.10. Let A be a Banach A-modules and let L be a closed ideal of A. If A is module Johnsonamenable, then A/L is module Johnson amenable.

Let A be a Banach A-module and let B be a Banach B-module. One can see that the Banach algebraA⊗B is a Banach A⊗B-module with following actions:

(α⊗ β)(a⊗ b) = α · a⊗ β · b (a, b ∈ A, α ∈ A, β ∈ B),

and similarity for the right action.

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proposition 2.11. Let A be a Banach A-module and let B be a Banach B-B-module. Suppose that B hasa non-zero idempotent. If A⊗B is module Johnson amenable (as Banach A⊗B-module), then A is moduleJohnson amenable.

The semigroup S is called inverse semigroup, if for every s ∈ S there exists an element s∗ ∈ S suchthat s = ss∗s and s∗ = s∗ss∗ [7]. Let S be an inverse semigroup with idempotents E. Consider `1(S) as aBanach module over `1(E) with the multiplication right action and the trivial left action, that is

δe · δs = δs, δs · δe = δse (s ∈ S, e ∈ E).

theorem 2.12. With the above notation, `1(S) is module Johnson amenable if and only if S is amenable.

3 Examples and applications

example 3.1. Let S be the set of natural numbers N with with maximum as its product. It is clear that Sis an inverse semigroup. Since S be an amenable group, Theorem 2.12 implies that `1(S) is module Johnsonamenable as an `1(E)-module, where E(S) = N. But it is not Johnson pseudo-contractible [3, Example 2.5].

example 3.2. Let S be the bicyclic semigroup. Then S is generated by p and q subject to qp = e forunit element e, that is S = pmqn : m,n ≥ 0. Following [6], S is an inverse amenable semigroup and(pmqn)∗ = pnqm. Theorem 2.12 implies that `1(S) is module Johnson amenable as an `1(E)-module, whereE(S) = pnqn : n ≥ 0. But it is not Johnson pseudo-contractible [3, Example 2.2].

example 3.3. Let G be an amenable group, I be an infinite set and let S = M0(G, I) be a Brandtsemigroup, that is the collection of all I × I matrices (g)i,j with g ∈ G in the (i, j)-th position and zeroelsewhere with the following multiplication

(g)i,j(h)k,l =

(gh)il if j = k,0 if j 6= k,

where g, h ∈ G and i, j, k, l ∈ I. S is an inverse amenable semigroup [6]. Theorem 2.12 implies that `1(S)is module Johnson amenable as an `1(E)-module, where E(S) = (e)ii : i ∈ I ∪ 0. But it is not Johnsonpseudo-contractible [10, Theorem 2.4].

The Banach algebra of I × I-matrices over C, with finite `1-norm and matrix multiplication is denoted

by MI(C), where I is an arbitrary set. Suppose that A =

[ai,j ] ∈ MI(C) | ∀i 6= j, ai,j = 0

as a

closed subalgebra of MI(C). One can see that A = MI(C) is Banach A-bimodule with respect to matrixmultiplication. Since a(α · b) = (a · α)b for every α ∈ A and a, b ∈ A, J = 0. So AJ = A and A∗∗

J⊥⊥ = A∗∗.

theorem 3.4. With above notation, let I be a non-empty set. Then MI(C) is module Johnson amenableif and only if I is finite.

References

[1] M. Amini, Module amenability for semigroup algebras, Semigroup Forum. 69 (2004), 243-254.

[2] M. Amini, A. Bodaghi, and D. Ebrahimi Bagha, Module amenability of the second dual and moduletopological center of semigroup algebras, Semigroup Forum. 80 (2010), 302-312.

[3] M. Askari, A. Pourabbas, and A. Sahami, Johnson pseudo-contractibility of certain Banach algebras andtheir nilpotent ideals, Anal. math. to appear (2018).

[4] A. Bodaghi, M. Amini, Module character amenability of Banach algebras, Arch. Math. 99 (2012),353-365.

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[5] A. Bodaghi, A. Jabbari, Module pseudo-amenability of Banach algebras, An. Stiint. Univ. Al. I. CuzaIasi Mat. (N.S.), TOM LXIII, f.3 (2017), 449-462.

[6] J. Duncan, I. Namioka, Amenability of inverse semigroups and their semigroup algebras, Proc. R. Soc.Edinb. 80A (1978), 309-321.

[7] J. M. Howie, An introduction to the theory of semigroups, London Math. Soc. Monogr, London-NewYork-San Francisco, 1976.

[8] H. Pourmahmood-Aghababa, and A. Bodaghi, Module approximate amenability of Banach algebras,Bull. Iranian Math. Soc. 39, no. 6 (2013), 1137-1158.

[9] H. Pourmahmood-Aghababa, (Super) module amenability, module topological centre and semigroupalgebras, Semigroup Forum. 81 (2010), 344-356.

[10] A. Sahami, A. Pourabbas, Johnson pseudo-contractibility of certain semigroup algebras, SemigroupForum. 97, Iss. 2 (2018), 203-213.

[11] A. Sahami, A. Pourabbas, Johnson pseudo-contractibility of various classes of Banach algebras, Bull.Belg. Math. Soc. Simon Stevin. 25 (2018), 171-182.

162

A Banach algebra associated with a locally compact groupoid

A. Shirinkalam*

(Islamic Azad University, Central Tehran Branch, Tehran, Iran)

Abstract: Let G be a locally compact groupoid with a fixed Haar system λ and a quasi invariantmeasure µ. We introduce the notion of λ-measurability and we put a new norm on Cc(G) the space ofcontinuous functions on G with compact support to make it a Banach algebra denoting by L1(G,λ, µ) andwe show that it is a two sided ideal of the algebra M(G) of complex Radon measures on G.

keywords. groupoid, λ-measurability, Haar system, L1-algebra.

subject. 22A22, 22A25

1 Introduction

For a locally compact group G with a Haar measure λ, the Banach algebra L1(G,λ) plays a central rolein harmonic analysis on G. This motivated us to define a similar notion in the case where G is a locallycompact groupoid. Following [1], in this article we give a new norm on Cc(G) and we denote by L1(G,λ, µ)its completion with respect to this norm. We show that the Banach algebra L1(G,λ, µ) plays a similar roleto the group algebra when we replace groups with groupoids.

For a locally compact groupoid G with a Haar system λ = λuu∈G0 and a quasi invariant measure µ,P. Hahn [5] introduced a norm on Cc(G) defined by ‖f‖I = max(‖f‖I,r, ‖f‖I,s) where

‖f‖I,r = supu(

∫|f |dλu) , ‖f‖I,s = supu(

∫|f |dλu).

Later Buneci [2, 3] give an equivalent norm with ‖.‖I as follows;

‖f‖I,µ = max

‖u→

∫|f | dλu‖∞, ‖u→

∫|f∗| dλu‖∞

.

She also defined another norm by ‖f‖II = supµ‖f‖II,µ where

‖f‖II,µ = sup

∫|f(x)j(s(x))k(r(x))|dν0(x);

∫|j|2dµ =

∫|k|2dµ = 1

.

Here f∗(x) = f(x−1). These norms are used to study of representations of Cc(G).

One may expect that as the group case, there is a full interaction between the properties of G andthat of L1(G,λ, µ). This is not completely true. For instance, unlike the group case, not every non-degenerate representation of L1(G,λ, µ) is integrated form of a representation of G. In section 2 we introducethe appropriate measurability notion used to define L1(G,λ, µ) and we introduce the algebra L1(G,λ, µ).

163

A. Shirinkalam

Sections 3 is devoted to the algebra structure of L1(G,λ, µ) and its embedding into the measure algebraM(G) as a closed ideal.

We start with some basic definitions. Our main reference for groupoids is [6].

A groupoid is a set G endowed with a product map: G2 → G; (x, y) 7→ xy, where G2 is a subsetof G × G called the set of composable pairs, and an inverse map: G → G; x 7→ x−1, such that for eachx, y, z ∈ G,

(i) (x−1)−1 = x,

(ii) if (x, y), (y, z) ∈ G2 then (xy, z), (x, yz) ∈ G2 and (xy)z = x(yz),

(iii) (x−1, x) ∈ G2 and if (x, y) ∈ G2 then x−1(xy) = y,

(iv) (x, x−1) ∈ G2 and if (z, x) ∈ G2 then (zx)x−1 = z.

If x ∈ G, s(x) = x−1x is called the source of x and r(x) = xx−1 is called the range of x. The pair (x, y)is composable if and only if s(x) = r(y). The set G0 = s(G) = r(G) is the unit space of G and its elementsare called units in the sense that xs(x) = x and r(x)x = x.

For u, v,∈ G0, Gu = r−1 u , Gv = s−1 v , Guv = Gu ∩ Gv and G u = Guu. The latter is a group,called the isotropy group at u. We put u ∼ v if Guv 6= ∅. Obviously ∼ is an equivalence relation on the unitspace G0. Its equivalence classes are called orbits, the orbit of u is denoted by [u].

A topological groupoid consists of a groupoid G and a topology compatible with the groupoid structure,such that the composition map is continuous on G2 in the induced product topology, and the inversion mapis continuous on G. A locally compact groupoid is a topological groupoid G which satisfies the followingconditions

(i) G0 is locally compact and Hausdorff in the relative topology.

(ii) There is a countable family C of compact Hausdorff subsets of G whose interiors form a basis forthe topology of G.

(iii) Every Gu is locally compact Hausdorff in the relative topology.

A locally compact groupoid is r-discrete if its unit space is an open subset. Let G be a locally compactgroupoid. The support of a function f : G → C is the complement of the union of all open, Hausdorffsubsets of G on which f vanishes. The space Cc(G) consists of all continuous functions on G with compactsupport. A left Haar system for G consists of measures

λu, u ∈ G0

on G such that

(i) the support of λu is Gu,

(ii) (continuity) for each f ∈ Cc(G), u 7→ λ(f)(u) =∫fdλu is continuous,

(iii) (left invariance) for any x ∈ G and any f ∈ Cc(G),

∫f(xy)dλs(x)(y) =

∫f(y)dλr(x)(y).

Let µ be a measure on G0. The measure on G induced by µ is ν =∫λud(µ), defined by

∫G fd(ν) =∫

G0dµ(u)

∫G fdλ

u, for f ∈ Cc(G). The measure on G2 induced by µ is ν2 =∫λu×λudµ(u). The inversion of

ν is ν−1 =∫λudµ(u). Note that the measures ν, ν2, ν−1 are Radon. A measure µ on G0 is said to be quasi-

invariant if its induced measure ν is equivalent to its inverse ν−1. Let µ be a quasi-invariant measure on G0.The Radon-Nikodym derivative D = dν

dν−1 is called the modular function of µ. An equivalent definition ofmodular function on G is given in [7, Definition 2.3], defining the modular function ∆ as a strictly positivecontinuous homomorphism on G such that ∆|Guu is modular function for Guu.

A subset A of G is called a G-set if the restrictions of r and s to it are one-to-one. Equivalently, A is aG-set if and only if AA−1 and A−1A are contained in G0 .

We give some notations from references [2, 3] which is related to the representations of L1(G,λ, µ). Letλuu be a fixed Haar system on G. Let µ be a quasi-invariant measure, ∆ its modular function, ν be the

measure induced by µ on G and ν0 = ∆−12 ν. Let

I(G, ν, µ) =f ∈ L1(G, ν0) : ‖f‖I,µ <∞

and

IIµ(G, ν, µ) =f ∈ L1(G, ν0) : ‖f‖II,µ <∞

164


where ‖f‖II,µ is defined by If µ1 and µ2 are two equivalent quasi-invariant measures, then

‖f‖II,µ1 = ‖f‖II,µ2 ,

because ‖f‖II,µ = ‖IIµ(|f |)‖ for each quasi-invariant measure µ, where IIµ is the one-dimensional trivialrepresentation on µ. Now define The supremum is taken over the class of quasi-invariant measures.

In this paper we frequently use the following version of Fubini’s theorem for (not necessarily σ-finite)Radon measures [4, Theorem B.3.3].

2 The algebra L1(G, λ, µ)

For the rest of the paper, G is a locally compact, Hausdorff, second countable groupoid which admits a leftHaar system λ = λu. First we have a definition.

definition 2.1. A Borel measurable set E ⊆ G is called λ-measurable if for each u ∈ G0, E ∩ Gu belongsto the σ-algebra Mλu . A function f : G → C is λ-measurable if for every u ∈ G0 and every open setO ⊆ C, f−1(O) ∩Gu ∈Mλu .

lemma 2.2. For each f : G→ C, λ-measurability of f is equivalent to ν-measurability of f .

Proof. We have suppλu = Gu and

f−1(O) = (f−1(O) ∩Gu) ∪ (f−1(O) ∩ (Gu)c).

Since λu is complete and (Gu)c is λu-null, (f−1(O) ∩ (Gu)c) ⊆ (Gu)c is in Mλu . Thus for each u ∈ G0 andopen set O ⊆ C

f−1(O) ∈Mλu ⇐⇒ f−1(O) ∩Gu ∈Mλu .

Now for ν =∫λudµ(u), we have Mν =

⋂u∈G0 Mλu , hence f is ν-measurable if and only if f is λ-measurable.

If λ =λu

is the completion of λ = λu and f : G → [0,∞] is λ-measureable, then there is a λ-measurable function g such that f = g on Gu (λu-a.e). To see this, let f be λu-measurable on Gu. Since λu isthe completion of λu, there exists a λu-measurable function gu such that f = gu (λu−a.e). Now define g = guon Gu and zero elsewhere. Since for every u ∈ G0 and every open set O ⊆ C, g−1(O)∩Gu = g−1

u (O) ∈Mλu ,g is λ-measurable and since λu(Gu)c = 0 we have f = g on Gu (λu-a.e). So from now on we assume thatthe Haar system λ is complete.

If f : G → C is λ-measurable and g : C → C is continuous, then g f : G → C is λ-measurable.Also, if f, g : G → R is λ-measurable, Φ : C → Y is continuous and h(x) = Φ(f(x), g(x)), then h is λ-measurable. If f = u+ iv, then f is λ-measurable if and only if u, v are λ-measurable. If f, g : G → R areλ-measurable, so are f + g and fg. Also, if fj∞1 is a sequence of R-valued λ-measurable functions, thenthe functions g1(x) = supj fj(x), g2(x) = infj fj(x), g3(x) = lim supj fj(x), and g4(x) = lim infj fj(x) areall λ-measurable. If f(x) = limj fj(x) exists for every x ∈ G, then f is λ-measurable. Thus if fj∞1 is asequence of complex-valued λ-measurable functions and fj → f λu-a.e, then f is λ-measurable.

Here is our main definition.

definition 2.3. Suppose µ is a quasi-invariant probability measure on G0 and ν is Radon measure inducedby µ. We define

L1(G, ν) = L1(G,λ, µ) =

f : G→ C : f is λ-measurable, ‖f‖1 =

∫

G|f(x)|dν(x) <∞

,

with the product given by (f ∗ g)(x) =∫Gr(x) f(y)g(y−1x)dν(y).

165

A. Shirinkalam

If f, g ∈ L1(G,λ, µ), then

‖f ∗ g‖1 =

∫

G|f ∗ g(x)|dν(x) ≤

∫

G

∫

Gr(x)

|f(y)||g(y−1x)|dν(y)dν(x)

≤∫

G

∫

G|f(y)||g(y−1x)|dν(y)dν(x) =

∫

G

∫

G|f(y)||g(x)|dν(y)dν(yx)

=

∫

G0

∫

G|g(x)|

∫

G0

∫

G|f(y)|dλu(y)dµ(u)dλu(yx)dµ(u) =

∫

G|g(x)|(

∫

G|f(y)|dν(y))dν(x)

= ‖f‖1‖g‖1.

Also the measurability of f ∗ g follows from λ-measurability of f, g.Next we define an involution on L1(G,λ, µ). We say that the assertion P (x) holds for λ-a.e.x if for

E = x : ¬P (x), µ u : λu(E) > 0 = 0. Clearly an assertion holds λ-almost everywhere if and only if itholds ν-almost everywhere.

lemma 2.4. Suppose Du : Gu → R+ with Du(x) = dλu(x)dλu(x) (x ∈ G). Then D = Du on Gu (a.e.).

Proof. Suppose E ⊆ Gu. We have

ν(E) =

∫

G0

λu(E)dµ(u) =

∫

G0

∫

Edλu(x)dµ(u) =

∫

G0

∫

EDu(x)dλu(x)dµ(u).

Also from dν = Ddν−1 we have

ν(E) =

∫

ED(x)dν−1(x) =

∫

ED(x)

∫

G0

dλu(x)dµ(u) =

∫

G0

∫

ED(x)dλu(x)dµ(u).

Thus ∫

G0

∫

E(Du(x)−D(x))dλu(x)dµ(u) = 0. (60)

Now let Eu = x ∈ Gu : Du(x) > D(x). If λu(Eu) > 0, then

∫

Eu(Du(x)−D(x))dλu(x) > 0.

But∫Eu(Du(x)−D(x))dλu(x) = 0 (µ-a.e.), hence λu(Eu) = 0 (µ-a.e.). Thus µ u : λu(Eu) > 0 = 0 so that

Du(x) ≤ D(x)(λ-a.e.). A similar argument leads to Du(x) ≥ D(x) (λ-a.e.).

The map ∗ : L1(G,λ, µ) → L1(G,λ, µ); f 7−→ f∗, where f∗(x) = f(x−1)D(x−1), is an isometricinvolution on L1(G,λ, µ). Note that from [3, page 9] we have

‖f‖L1(G,λ,µ) = ‖f∗‖L1(G,λ,µ) = ‖f‖L1(G,ν0) = ‖f∗‖L1(G,ν0) ≤ ‖f‖II,µ = ‖f∗‖II,µ ≤ ‖f‖I,µ = ‖f∗‖I,µ.

Hence the space of L1(G,λ, µ) is in general bigger than I(G, ν, µ) and IIµ(G, ν, µ) with respect to I-normand II-norm, indeed I(G, ν, µ) ⊆ IIµ(G) ⊆ L1(G,λ, µ).

By [3, page 15] the space of continuous functions with compact support Cc(G) has a two-sided boundedapproximate identity. Since Cc(G) is dense in L1(G,λ, µ), thus L1(G,λ, µ) has a two-sided bounded approx-imate identity.

For each f ∈ L1(G,λ, µ) define

Lxf(y) = f(x−1y), Rxf(y) = f(yx),

when the multiplications on the right hand sides are defined. It is easy to check that the maps Lx, Rx arehomomorphisms.

lemma 2.5. Let I be a closed subspace of L1(G,λ, µ). Then I is a left ideal if and only if it is closed underleft translation, and I is a right ideal if and only if it is closed under right translation.

166


Proof. Note that since f ∗ g =∫Gr(y) f(y)Lyg dν(y),

Lx(f ∗ g) =

∫

Gr(y)

f(y)LxLyg dν(y) =

∫

Gr(y)

f(y)Lxyg dν(y)

=

∫

Gr(y)

f(x−1y)Lyg dν(y) =

∫

Gr(y)

Lxf(y)Lyg dν(y) = (Lxf) ∗ g.

Now suppose (en)n is a bounded approximate identity for L1(G,λ, µ). For the first assertion, if f ∈L1(G,λ, µ) and g ∈ I and I is a left ideal, then we have

Lx(en) ∗ f = Lx(en ∗ f) −→ Lxf.

Conversely, if I is closed under left translation and f ∈ L1(G,λ, µ) and g ∈ I,

f ∗ g =

∫

Gr(y)

f(y)Lyg dν(y)

is in the closed linear span of the functions Lyg and hence in I. The other assertion is proved similarly.

3 The involutive Banach algebra M(G)

In this section we show that L1(G,λ, µ) is a closed ideal in the algebra of complex Radon measures onG. Let M(G) be the space of complex Radon measures on G. If η, θ ∈ M(G), then the map ψ 7→ I(ψ)on C0(G) defined by I(ψ) =

∫G

∫Gr(y) ψ(xy)dη(x)dθ(y) is a linear functional on C0(G) satisfying |I(ψ)| ≤

‖ψ‖sup‖η‖‖θ‖, so by Riesz representation theorem, it is given by a measure shown as η ∗ θ called the

convolution of η, θ with ‖η ∗ θ‖ ≤ ‖η‖‖θ‖. If we define η∗(E) = η(E−1) then η 7→ η∗ is an involution onM(G), and M(G) is a Banach ∗-algebra. In this section we show that the space L1(G,λ, µ) is a closedtwo-sided ideal of M(G).

proposition 3.1. The map L1(G,λ, µ) → M(G); f 7→ νf defined by νf (E) =∫E fχE dν (E ⊆ G) is an

isometric embedding.

Proof. If f ∈ L1(G,λ, µ), then f is λ-measurable so the integral exists and it is easy to check that νfis a measure on G. We show that νf is Radon. If f = u + iv then νf = νu + iνv, so νf is Radon ifand only if νu and νv are Radon. Since G is locally compact Hausdorff and second countable, we haveνu(K) =

∫K u dν ≤

∫G |u|dν = ‖u‖1 <∞, for each compact set K, thus νu is Radon. Similarly νv is Radon,

and so is νf .By definition, ‖νf‖ = sup ∑n

1 |νf (Ei)| : n ∈ N,G =⊔n

1 Ei, so for each ε > 0 there exists a partitionEin1 of G such that

‖νf‖ − ε <n∑

1

|νf (Ei)| =n∑

1

|∫

GfχEidν| ≤

∫

G(|f |

n∑

1

χEi)dν = ‖f‖1.

Thus ‖νf‖ ≤ ‖f‖1. Conversely, suppose f ≥ 0 then νf ≥ 0 and for every partition Ein1 of G we have,

‖νf‖ ≥n∑

1

νf (Ei) =n∑

1

∫

GfχEi dν =

∫

Gfdν = ‖f‖1.

If f = u + iv = (f1 − f2) + i(f3 − f4), where fi ≥ 0 then ‖νf‖ = ‖νf1‖ + ‖νf2‖ + ‖νf3‖ + ‖νf4‖ ≥‖f1‖1 + ‖f2‖1 + ‖f3‖1 + ‖f4‖1 ≥ ‖f‖1. Hence ‖νf‖ ≥ ‖f‖1 and equality holds.

The above proposition shows that L1(G,λ, µ) is a closed subspace of M(G). Next we show that it isindeed an ideal.

lemma 3.2. If f, g ∈ L1(G,λ, µ), then ν(f∗g) = νf ∗ νg.

167

A. Shirinkalam

Proof. For each compact set K we have

νf ∗ νg(K) =

∫

GχK(x)d(νf ∗ νg)(x) =

∫

G

∫

Gr(x)

χK(yx)dνf (y)dνg(x)

=

∫

Gf(y)

∫

Gs(y)

χK(yx)dνg(x)dν(y) =

∫

G

∫

Gs(y)

f(y)g(y−1x)χK(x)dν(y)dν(x)

=

∫

GχK(x)

∫

Gr(x)

f(y)g(y−1x)dν(y)dν(x) =

∫

G(f ∗ g)(x)χK(x) dν(x) = νf∗g(K).

Since νf∗g and νf ∗νg are regular measures, the equality holds for each open set and then for each measurableset.

If f ∈ L1(G,λ, µ) and η ∈ M(G), we shall define η ∗ f in such a way that νη∗f = η ∗ νf . Supposeϕ ∈ C0(G) and put

ν(η ∗ f)(ϕ) =

∫

Gϕ(x) dνη∗f (x) =

∫

Gϕ(x)(η ∗ f)(x) dν(x).

On the other hand,

η ∗ νf (ϕ) =

∫

G

∫

Gr(x)

ϕ(yx)dη(y)dνf (x) =

∫

G

∫

Gr(x)

ϕ(yx)dη(y)f(x)dν(x)

=

∫

G

∫

Gs(y)

f(y−1x)ϕ(x)dη(y)dν(y−1x) =

∫

Gϕ(x)

∫

Gs(y)

f(y−1x)dη(y)dν(x).

Comparing these equalities implies that

(η ∗ f)(x) =

∫

Gr(x)

f(y−1x)dη(y).

If f ∈ L1(G,λ, µ), then it is easy to check that

∫

GuRyf(x)dλu(x) =

∫

Guf(x)dλu(xy−1) = D(y−1)

∫

Guf(x)dλu(x).

Thus ∫

GRyf(x)dν(x) = D(y−1)

∫

Gf(x)dν(x).

Similarly, we want to define f ∗ η in such a way that the equality ν(f∗η) = νf ∗ η holds. Again supposeϕ ∈ C0(G). We have

ν(f∗η)(ϕ) =

∫

Gϕ(x) dνf∗η(x) =

∫

Gϕ(x)(f ∗ η)(x) dν(x) =

∫

G0

∫

Gϕ(x)(f ∗ η)(x) dλu(x)dµ(u)

=

∫

G0

∫

Guϕ(x)(f ∗ η)(x) dλu(x)dµ(u) =

∫

Guϕ(x)(f ∗ η)(x) dν(x).

On the other hand,

(νf ∗ η)(ϕ) =

∫

G

∫

Gr(y)

ϕ(xy)dνf (x)dη(y) =

∫

G

∫

Gr(y)

ϕ(xy)f(x)dν(x)dη(y)

=

∫

G

∫

Gr(y)

ϕ(x)f(xy−1)dν(xy−1)dη(y) =

∫

G

∫

Gr(y)

ϕ(x)f(xy−1)D(y−1)dν(x)dη(y)

=

∫

Gs(x)

ϕ(x)

∫

Gf(xy−1)D(y−1)dη(y)dν(x).

Comparing the above equalities, we have

(f ∗ η)(x) =

∫

Gf(xy−1)D(y−1)dη(y).

168


corollary 3.3. L1(G,λ, µ) is a two sided closed ideal of M(G).

Proof. Suppose f ∈ L1(G,λ, µ) and η ∈M(G). Then we have

‖η ∗ f‖1 =

∫

G|η ∗ f(x)|dν(x) ≤

∫

G

∫

Gr(x)

|f(y−1x)| d|η|(y)dν(x)

=

∫

G

∫

Gr(x)

|f(x)| d|η|(y)dν(yx) =

∫

G|f(x)|

∫

Gr(x)

d|η|(y)dν(x)

≤ ‖η‖‖f‖1 <∞.

Thus η ∗ f ∈ L1(G,λ, µ). Also

‖f ∗ η‖1 =

∫

G|(f ∗ η)(x)|dν(x) ≤

∫

G

∫

G|f(xy−1)D(y−1)|d|η|(y)dν(x)

=

∫

G

∫

G|f(x)|D(y−1)D(y)dν(x)d|η|(y)

=

∫

G

∫

G|f |dνd|η| = ‖f‖1‖η‖ <∞.

Hence f ∗ η ∈ L1(G,λ, µ).

Acknowledgment

The author should thank to Professor Masoud Amini for his valuable helps and supports.

References

[1] M. Amini, A. Medghalchi and A. Shirinkalam, L1-Algebra of a locally compact groupoid, ISRN Algebra,2011, doi:10.5402/2011/856709.

[2] M. R. Buneci, Isomorphic groupoid C∗-algebras associated with different Haar systems, New York J.Math. 11 (2005), 225–245.

[3] M. R. Buneci, The equality of the reduced and the full C*-algebras and the amenability of a topologicalgroupoid, Oper. Theory Adv. Appl. 153 (2005), 61–78.

[4] A. Deitmar, S. Echterhoff, Principles of Harmonic Analysis, Springer, New York, 2009.

[5] P.Hahn, The regular representation of measure groupoids, Trans. Amer. Math. Soc., 519 (1978),35-72.

[6] J. Renault, A groupoid approach to C∗-algebras, Lecture Notes in Mathematics 793, Springer, Berlin,1980.

[7] J. Westman, Harmonic analysis on groupoids, Pacific J. Math. 27 (1968), 621-632.

169

Multipliers on A1(G/H) and Lp(G/H)

V. Yousefiazar* and M. H. Sattari

(Shahid Madani University, Tabriz, Iran)

Abstract:Let G be a compact group and H be a closed subgroup of G. In this paper we study multi-pliers on A1(G/H) and Lp(G/H) and prove the Wendels theorem for A1(G/H).

keywords. Banach algebra, homogeneous space, multiplier

subject. 32A65; 43A85; 42B15

1 Introduction

Throught this paper, G is a compact group and H is a compact subgroup of G and λG, λH are Haarmeasures on G,H respectively.The quotient space G/H is considered as a homogeneous space that G actson it by x(yH) = (xy)H. let µ be a Radon measure on G/H, for allx ∈ G, the measure µx is definedby µx(E) = µ(xE), where E is a Borel subset of G/H. µ is said to be G-invariant if µx = µ for allx ∈ G. Let µ is a G-invariant measure on G/H and(1 ≤ p ≤ ∞), Lp(G/H) = Lp(G/H,µ). The mapTH : Lp(G) −→ Lp((G/H), µ) is defined by THf(xH) =

∫H f(xξ)dλH(ξ) (xH ∈ G/H). Lp(G/H) is a

Banach algebra by with the product

ϕ ? ψ = TH(ψ q ? ψ q) (ϕ,ψ ∈ Lp(G/H)).

The map T : M(G) −→ M(G/H) was introduced by Reiter and Stageman in [4], where for every m ∈M(G), T (m)(E) = m(q−1(E)) (E ⊆ G/H is a Borel set). M(G/H) is a Banach algebra. L1(G/H)is an ideal of M(G/H) [3], and for ϕ ∈ L1(G/H) and ν ∈ M(G/H), x ∈ G, then (ϕ ? ν)(xH) =∫GH

∫H ϕ(xξy−1H)dλH(ξ)dν(yH),

(ν ? ϕ)(xH) =∫GH

∫H ϕ(ξy−1xH)dλH(ξ)dν(yH).

Define A1(G/H) = ϕ ∈ L1(G/H), Lh(ϕ) = ϕ∀h ∈ H. A1(G/H) is a Banach subalgebra of L1(G/H). LetM1(G/H) be the linear subspace of M(G/H) given by

M1(G/H) = ν ∈M(G/H) : Lhν = ν, ∀h ∈ H,

M1(G/H) is a Banach subalgebra of M(G/H). For mor details see [2]

definition 1.1. Let A be a Banach algebra. A linear operator T is a left (resp. right) multiplier if for everya, b ∈ A, T (ab) = T (a)b (resp. T (ab) = aT (b)). A multiplier is a pair (L,R) , where L and R are left andright multipliers on A, respectively, and aL(b) = R(a)b. The set of left multipliers, right multipliers, andmultipliers on A are denoted by Ml(A),Mr(A), and M(A), respectively.

170

Multipliers on A1(G/H) and Lp(G/H)

2 Main results

theorem 2.1. Let G be a compact group and H be a closed subgroup of G. If 1 ≤ p < ∞ and q be aconjuate exponent to p, then:(i) Mr(L

p(G/H), C(G/H)) ∼= Lq(G/H)(ii) Mr(L

p(G/H), L∞(G/H)) ∼= Lq(G/H)(iii) Mr(C(G/H), C(G/H)) ∼= M(G/H)(iv) Mr(C(G/H), L∞(G/H)) ∼= M(G/H)

Proof. (i) Since G is compact, by Theorem 4.13 [2] Lp(G/H) is a Banach algebra. Let ϕ ∈ C(G/H) andψ ∈ Lp(G/H), so that ϕ ? ψ = TH(ϕ q ? ψ q) ∈ C(G/H) and ψ ? ϕ = TH(ψ q ? ϕ q) ∈ C(G/H) and

‖ϕ ? ψ‖∞ = ‖TH(ϕ q ? ψ q)‖∞≤ ‖ϕ q ? ψ q)‖∞≤ ‖ϕ q‖∞‖ψ q‖p= ‖ϕ‖∞‖ψ‖p

Thus C(G/H) is a Lp(G/H)-bimodule, and so M(G/H) is a Lp(G/H)-bimodule.Let T ∈Mulr(L

p(G/H), C(G/H)). The operator T ? : M(G/H) −→ Lq(G/H) is a left multiplier. We knowM(G/H) has a right unital, set T ∗(eG/H) = ψ0. Therefore T ∗(ϕ) = ϕ ? ψ0 and T (ϕ) = ϕ ? (TH( ˇ(ψ0 q))),where ˇ(ψ0 q)(x) = (ψ0 q)(x−1)(ii) Let (eα) be a right approximate identity for Lp(G/H) and T ∈Mulr(L

p(G/H), L∞(G/H))

T (f) = limαT (f ? eα) = lim

αf ? T (eα) ∈ C(G/H),

the result follows from(i).The proof of (iii) is similar to (i). (iv) can be concluded from (i) and (iii) respectively.

theorem 2.2. Let G be a compact group and H be a closed subgroup of G. If T : A1(G/H) −→ A1(G/H)be a bounded linear operator such that T (ϕ ? ψ) = ϕ ? T (ψ) (or T (ϕ ? ψ) = T (ϕ) ? ψ)), then there existsµ ∈ M1(G/H) such that T (ϕ) = ϕ ? µ (T (ϕ) = µ ? T (ϕ)).

Proof. The result follows by Proposition 2.1.3 [5].

theorem 2.3. Let G be a compact group and H be a closed subgroup of G. For µ ∈M1(G/H) define thelinear bounded operator Lµ : A1(G/H) −→ A1(G/H) by Lµ(f) = µ ? f , then ‖Lµ‖ = ‖µ‖.

theorem 2.4. Let G be a compact group and H be a closed subgroup of G. Then M(A1(G/H)) isisometrically isomorphic to M1(G/H).

Proof. The result follows by Theorem 2.3 and Proposition 1.4.26 [1]

References

[1] H.G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs 24Clarendon Press, Oxford, 2000.

[2] A. G. Farashahi, Abstract measure algebras over homogeneous spaces of compact groups, InternationalJournal of Mathematics, 1850005 (2018), (34 pages)

[3] H. Javanshiri, N. Tavallaei, Measure algebras on homogeneous spaces, arxiv:1606.08773vl (math.CA),28JUN (2016), pp.1-18.

[4] H.Reiter , D.Stegeman, Classical Harmonic Analysis and locally compact group, oxford uni press, 1968.

[5] V.Runde, Lecture Note in Mathematics, Springer Berlin, 2004.

171

Orlicz algebra Lϕ(S)

V. Yousefiazar*

(Shahid Madani University, Tabriz, Iran)

Abstract: we are going to study some conditions on ϕ (ϕ is a young function) that `ϕ(S) become anorlicz algebra, when S is a discrete semigroup. Also we investigated amenability and weak amenability `ϕ(S).

keywords. Amenable, Orlicz space , Semigroup algebra (between 3 to 5 keywords)

subject. 43A07,43A10,46E30

1 Introduction

A function ϕ : R −→ [0; 1] is called a Young function if ϕ is a convex, even, and left continuous functionwith ϕ(0) = 0 which is neither identically zero nor identically infinite. For any Young function ϕ let

ψ(x) = supxy − ϕ(y) : y ∈ R (x ∈ R)

It is easily verified that ψ is a Young function called the complementary Young function to ϕ. It should beremarked that ϕ is also the complementary Young function to ψ.The (ϕ;ψ) is called a complementary pair of Young functions. let L0(Ω) denote the set of all equivalenceclasses of measurable complex-valued functions on Ω -measurable complex-valued functions on Ω. For f ∈ Ωdefine

ρϕ(f) =

∫

Ωϕ(|f(x)|)dx.

Then the Orlicz spaceLϕ(Ω) is defined by

Lϕ(Ω) = f ∈ L0(Ω) : ρϕ(f) <∞

We also setMϕ(Ω) = f ∈ L0(Ω) : ρϕ(αf) <∞,∀α > 0

Then Lϕ(Ω) is a Banach space under the norm Nϕ(Ω), called the Luxemburg-Nakano norm, defined forf ∈ Lϕ(Ω) by

Nϕ(f) = infK : ρϕ(f/K) ≤ 1It is well known that Nϕ(f) ≤ 1 if and only if ρϕ(f) ≤ 1. Furthermore, if the Young function ϕ is strictlyincreasing and continuous, then using the complementary Young function ψ another norm ‖.‖ϕ, called theorlicz norm, is defined on Lϕ(Ω) in the following way :

‖f‖ϕ = sup∫

Ω|fg|dx : ρψ(g) ≤ 1

172


Let us remark that ‖.‖ϕ is equivalent to Nϕ(.); in fact, Nϕ(f) ≤ ‖f‖ϕ ≤ 2Nϕ(f), for all f ∈ Lϕ(Ω). LetSϕ(Ω) be the closure of the linear space of all step functions in Lϕ(Ω). Then Sϕ(Ω) is a Banach space andcontains Cc(G)For 1 ≤ p ≤ ∞, classical Lebesgue spaces on G , (G is a locally compact group ) with respect to the Haarmeasure λ will be denoted by Lp(G) with the norm ‖.‖p. It is clear that Lp(G) is an elementary example ofthe orlicz space Lϕ(G). For more details see [3]

definition 1.1. Let S a semigroup. The semigroup algebra `1(S) is the completion in the `1- norm of thealgebra CS ( for a semigroup S, we denote by CS the dense subalgebra of finitly-supported elements ). Theconvolution product ? 0n `1(S) is defined by requiring that

δs ? δt = δst (s, t ∈ S)

Forf, g ∈ `1(S) we have

h ? g(r) =∑

st=r

f(s)g(t) (s, t ∈ S)

where the sum is defined to be zero if (s, t) : st = r = ∅. We define

`ϕ(S) = f ∈ `1(S) : ρϕ(f) <∞

2 Main results

theorem 2.1. Suppose ϕ1, ϕ2, ϕ3 are young functions which satisfy, for x ≥ 0

ϕ−11 (x)ϕ−1

2 (x) ≤ xϕ−13 (x)

and that f ∈ Lϕ1(S), g ∈ Lϕ2(S) on a semigroup S. Then their convolution

(h(x) =∑

x=rs

f(r)g(s) (s, r ∈ S)

is in Lϕ3(S) and

‖h‖ϕ3 ≤ 2‖f‖ϕ1‖g‖ϕ2

Proof. Let ε > 0. Without loss of generality we may suppose ‖f‖ϕ1 = ‖g‖ϕ2 = 1 by Lemma 2.4 [1] andjensen’s inequality

∑

x∈Sϕ3(

|h(x)|2(1 + ε)2

) ≤∑

x∈Sϕ3(1/2

∑

rs=x

|f(r)||g(s)|(1 + ε)2

)

≤∑

x∈Sϕ3[1/2

∑

rs

ϕ1(|f(r)|(1 + ε)

)ϕ−13 (ϕ2(

|g(s)|(1 + ε)

)) + 1/2∑

rs

ϕ2(|g(s)|1 + ε

)ϕ−13 (ϕ1(

|f(r)|1 + ε

)]

≤ 1/2∑

x∈Sϕ3[∑

x=rs

ϕ1(|f(r)|(1 + ε)

)ϕ−13 (ϕ2(

|g(s)|(1 + ε)

))]

+ 1/2∑

x∈Sϕ3[∑

x=rs

ϕ2(|g(s)|1 + ε

)ϕ−13 (ϕ1(

|f(r)|1 + ε

)]

= I + J,

173

V. Yousefiazar

we use Jensen’s inequality (10.8) and (10.1)[2], observing that∑ϕ1( |f(t)|

1+ε ) ≤ 1

I = 1/2∑

x∈Sϕ3[∑

x=rs

ϕ1(|f(r)|1 + ε

)ϕ−13 (ϕ2(

|g(s)|1 + ε

)]

≤ 1/2∑

x∈Sϕ3[(

∑

x=rs

ϕ1(|f(r)|1 + ε

)ϕ−13 (ϕ2(

|g(s)|1 + ε

)/∑

x=rs

(ϕ1(|f(r)|1 + ε

)]

≤ 1/2∑

x∈S((∑

x=rs

ϕ1(|f(r)|1 + ε

)ϕ−13 (ϕ3(ϕ2((

|g(s)|1 + ε

))/∑

x=rs

ϕ1(|f(r)|1 + ε

)

≤ 1

2∑

x=rs ϕ1( |f(r)|1+ε )

∑

x∈S((∑

x=rs

(ϕ1(|f(r)|1 + ε

)(ϕ2((|g(s)|1 + ε

)

=1

2∑

r∈S ϕ1( |f(r)|1+ε )

∑

r∈S((∑

x=rs

(ϕ1(|f(r)|1 + ε

)(ϕ2((|g(r−1x)|

1 + ε)

≤ 1

2∑

r∈S ϕ1( |f(r)|1+ε )

(∑

r∈Sϕ1(|f(r)|1 + ε

)(∑

x=rs

ϕ2(|g(r−1x)|

1 + ε))

≤ 1

2∑

r∈S ϕ1( |f(r)|1+ε )

(∑

r∈Sϕ1(|f(r)|1 + ε

)

= 1/2.

Similarly J ≤ 1/2 and I + J ≤ 1.Thus ‖h‖ϕ3 ≤ 2(1 + ε)2. the theorem follows by ε −→ 0.

theorem 2.2. Let ϕ be a young function and S be a semigroup. If ϕ′(0) > 0 or S is compact, then `ϕ(S)is a Banach algebra under convolution.

Proof. If S is compact, then S is finite. Thus `ϕ(S) ' `1(S). If ϕ′(0) > 0, then `ϕ(S) ⊂ `1(S).The resultfollows by Theorem 2.1 . It is enough to put there ϕ1(x) = ϕ3(x) = ϕ(x) and ϕ2(x) = |x|.

definition 2.3. A Banach algebra A is amenable if every bounded derivation from A into the dual of anyBanach A-bimodule E is inner.

definition 2.4. A Banach algebra A is weakly amenable if every bounded derivation from A into A′ inner.

definition 2.5. Let S be a semigroup. An element p ∈ S is an idempotent if p2 = p ;the set of idempotentsof S is denoted by E(S).

definition 2.6. Let S be a semigroup. For s ∈ S, define Ls and Rs by Ls(t) = st and Rs(t) = ts. An elements ∈ S is left (respectively, right) cancellable if Ls (respectively, Rs) is injective on S,and s is cancellable ifit is both left cancellable and right cancellable. The semigroup S is left (respectively, right) cancellative ifeach element is left (respectively, right) cancellable, and cancellative if each element is cancellable.

theorem 2.7. Let S be a semigroup. If S is abelian and E(S) = S and ϕ be a young function such thatϕ′(0) > 0, then `ϕ(S) is weakly amenable.

Proof. The result follows by Proposition 2.8.72 [2].

theorem 2.8. Let S be a left or right cancellative semigroup E(S) = S . Let ϕ be a young function suchthat ϕ′(0) > 0 and ϕ ∈ ∆2 . If `ϕ(S) is amenable then Sis finite.

Proof. If `ϕ(S) is amenably, then `ϕ(S) has a bounded approximate identity fi. Suppose that f ∈ `ϕ(S) =(Sψ(S))′ is a cluster point of fi. we have

δt ? f = δt

therefore f(t) = 1 for all t ∈ S. However, this is only possible if S is finite.

174


References

[1] R.O’Neil, Fractional integration in orlicz spacesI,Trans.of A.M.S.115(1965), 300-328.

[2] H. G. Dales and H. Dedania, Weighted convolution algebras on subsemigroups of the real line, Disserta-tiones Mathematicae (Rozprawy Matematyczne) 459 (2009), 160. MR2477218 [21] H. G. Dales and A.T.-M. Lau, The second duals of Beurling algebras, Memoirs American

[3] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces (Marcel Dekker, New York, 1991).

175

Banach space valued mappings with compact image

F. Zaj* and M. Abtahi

(Damghan University, Iran)

Abstract: Let Ω be a topological space and A be a Banach space. Then Ck(Ω, A) denotes the spaceof all continuous functions on Ω, such that the closure f(Ω) is compact in A. If Y is a compact Hausdorffspace, we show that Ck(Ω, C(Y )) is isometrically isomorphic with the injective tensor product Cb(Ω)⊗εC(Y ),where Cb(Ω) is the space of all bounded and continuous functions on Ω.

keywords. Commutative Banach algebras, Continuous functions, Mappings with compact image


1 Introduction

Let A be a Banach space, and Ω be a topological space. We denote by Cb(Ω, A) the space of all A-valuedbounded and continuous functions on Ω and by Ck(Ω, A) the subspace of Cb(Ω, A) consisting of continuousfunctions f : Ω −→ A such that f(Ω) is compact in A. If A has the Heine-Borel property (equivalently, ifA is finite dimensional) then Ck(Ω, A) = Cb(Ω, A).

The main purpose of this note is to investigate if Ck(Ω, A) is isometrically isomorphic with Cb(Ω)⊗εA.We show that, this is always the case if A = C(Y ) for some compact Hausdorff space Y . This is equivalentto showing that, for every ε ≥ 0, for every f ∈ Ck(Ω, A), there exist a1, . . . , an ∈ A and f1, . . . , fn ∈ Cb(Ω)such that

‖f −n∑

i=1

fiai‖Ω ≤ ε,

where ‖ · ‖Ω is the uniform norm on Cb(Ω, A).

2 Main Results

We start with a definition.

definition 2.1. Let Ω be a topological space and A be a Banach space. A function f : Ω → A is called amapping with compact image if the closure f(Ω) is compact in A. The space of all functions f ∈ C(Ω, A)with compact image is denoted by Ck(Ω, A).

The uniform norm on Cb(Ω, A) is defined as usual;

‖f‖Ω = sup‖f(x)‖ : x ∈ Ω (f ∈ Cb(Ω, A)).

If Ω is compact, then C(Ω, A) = Cb(Ω, A) = Ck(Ω, A).The following lemma is very important in our work. In the following, given f ∈ C(Ω, C(Y )), we write

f(x, y) instead of f(x)(y), for x ∈ Ω and y ∈ Y .

176


lemma 2.2. Let Ω be a topological space and Y be a compact Hausdorff space, and f ∈ Ck(Ω, C(Y )).Then the function F : Y → Cb(Ω), F (y) = fy is continuous, where fy(x) = f(x, y), for x ∈ Ω.

Proof. It is obvious that, for every x ∈ Ω, the function fy : Ω → C, fy(x) = f(x, y) is continuous, that isfy ∈ C(Ω). To show that F is continuous, it suffices to show that, for every ε > 0 and every y ∈ Y , there isa neighborhood Vy of y such that

|f(x, t)− f(x, y)| < ε (x ∈ Ω, t ∈ Vy).

For every x ∈ Ω, set Ux = B(f(x), ε), the open ball with center f(x) and radius ε. Then

f(Ω) ⊆⋃

x∈Ω

Ux.

It is easily seen that we also have

f(Ω) ⊆⋃

x∈Ω

Ux.

Since, by the assumption, f(Ω) is compact, there are x1, · · · , xn ∈ Ω such that f(Ω) ⊆ Ux1 ∪ · · · ∪Uxn .From continuity of f(xi), as a function on Y , we get a neighborhood Vy of y that

|f(xi, t)− f(xi, y)| ≤ ε (t ∈ V iy ).

Let Vy = V 1y ∩ · · · ∩ V n

y . If x ∈ Ω, then f(x) ∈ Uxi for some xi so that |f(x, s) − f(xi, s)| ≤ ε, for alls ∈ Y. Now, for every t ∈ Vy, we have

|f(x, t)− f(x, y)| ≤ |f(x, t)− f(xi, t)|+ |f(xi, t)− f(xi, y)|+ |f(xi, y)− f(x, y)|≤ 2‖f(x)− f(xi)‖Y + |f(xi, t)− f(xi, y)| ≤ 3ε.

The proof is complete if we replace ε withε

3.

The following is the main result of this note. The proof is elementary, based on 2.2.

theorem 2.3. Let Ω be a topological space and Y be a compact Hausdorff space. Then Ck(Ω, C(Y )) isisometrically isomorphic with the injective tensor product Cb(Ω)⊗εC(Y ).

Proof. The universal property of tensor products, allow us to identify any tensor element∑n

i=1 fi ⊗ gi inCb(Ω)⊗C(Y ) with the corresponding C(Y )-valued function

∑ni=1 figi in C(Ω, C(Y )). We only need to show

that set

f1g1 + · · ·+ fngn : n ∈ N, fi ∈ Cb(Ω), gi ∈ C(Y ), 1 ≤ i ≤ n

is dense in Ck(Ω, C(Y )).

Let f ∈ Ck(Ω, C(Y )). Let ε > 0. For every y ∈ Y , by Lemma 2.2, there is a neighborhood Vy of y suchthat

|f(x, t)− f(x, y)| ≤ ε (x ∈ Ω, t ∈ Vy).

Since Y is compact, there are y1, . . . , yn ∈ Y such that Y ⊆ ⋃ni=1 Vyi . Let g1, . . . , gn ⊂ C(Y ) be a partition

of unity on Y subordinate to Vy1 , . . . , Vyn; so that 0 ≤ gi ≤ 1,∑n

i=1 gi = 1 on Y and gi = 0 on V cyi .

177

F. Zaj, M. Abtahi

For every 1 ≤ i ≤ n, put fi(x) = f(x, yi), x ∈ Ω. Then fi ∈ Cb(Ω) and, for every x ∈ Ω, we have

∥∥∥f(x)−n∑

i=1

fi(x)gi

∥∥∥Y

= supy∈Y

∣∣∣f(x, y)−n∑

i=1

f(x, yi)gi(y)∣∣∣

= supy∈Y

∣∣∣f(x, y)

n∑

i=1

gi(y)−n∑

i=1

f(x, yi)gi(y)∣∣∣

= supy∈Y

∣∣∣n∑

i=1

gi(y)(f(x, y)− f(x, yi))∣∣∣

≤ supy∈Y

n∑

i=1

|f(x, y)− f(x, yi)|gi(y)

≤ ε supy∈Y

n∑

i=1

gi(y) ≤ ε

(61)

Hence∥∥f −∑n

i=1 figi∥∥

Ω≤ ε.

Let X be a compact space and let A be a commutative Banach algebra with unit element 1. A BanachA-valued function algebra on X is a subalgebra A of C(X,A) such that (1) A contains the constant functions,(2) A separates the points of X, and (3) A is endowed with some complete algebra norm ‖ · ‖, not less thanthe uniform norm, such that the restriction of ‖·‖ to A is equivalent to the original norm of A. The A-valuedfunction algebra A is called admissible if (φ f)1 ∈ A, for all φ ∈M(A) and f ∈ A. See [1] and [5] for moreon vector-valued function algebras.

For any C-valued function algebra A, the algebraic tensor product A⊗A is an A-valued function algebra.If there is an algebra cross-norm ‖ · ‖γ on A⊗A, then the completion A⊗γA is a Banach A-valued functionalgebra [3]. Hence, by Tomiyama’s theorem [4] the maximal ideal space M(A⊗γA) is homeomorphic withM(A)×M(A); see also [2]. Many well-known examples of Banach A-valued function algebras are presentedas tensor products. For example, C(X,A) = C(X)⊗εA, P (X,A) = P (X)⊗εA, and R(X,A) = R(X)⊗εA(for the latter see [1, Theorem 2.6]).

The following shows that any C(Y )-valued uniform algebra is of the A⊗εC(Y ).

theorem 2.4. Let X and Y be compact Hausdorff spaces, and let A be an admissible C(Y )-valued uniformalgebra on X. Then A is isometrically isomorphic with an injective tensor product A⊗εC(Y ).

Proof. Let A = φ f : φ ∈ M(A), f ∈ A. In fact A is the subalgebra of A consisting of scalar valuedfunctions. We show that A = A⊗εC(Y ). To this end, it suffices to show that the set

f1g1 + · · ·+ fngn : n ∈ N, fi ∈ A, gi ∈ C(Y ), 1 ≤ i ≤ n

is dense in A. Take f ∈ A. Since X is compact and f is continuous, the image f(X) is compact inC(Y ). Hence, Lemma 2.2 is applicable to f . Following the proof of Theorem 2.3, there are y1 . . . , yn ∈ Y ,g1, . . . , gn ⊂ C(Y ), and a finite open covering Vy1 , . . . , Vyn of Y such that 0 ≤ gi ≤ 1,

∑ni=1 gi = 1 on Y

and gi = 0 on V cyi . Take fi(x) = f(x, yi), x ∈ X. In fact, fi = εyi f , where εyi : C(Y ) → C, g 7→ g(yi), is

the evaluation homomorphism at yi. Since A is admissible fi = εyi f ∈ A. Same calculations, as in (61),shows that ∥∥∥f −

n∑

i=1

figi

∥∥∥X≤ ε.

The proof is complete.

References

[1] M. Abtahi, Vector-valued characters on vector-valued function algebras, Banach J. Math. Anal. 10(3)2016, 608–620.

178


[2] M. Abtahi, On the character space of Banach vector-valued function algebras, Bull. Iranian Math. Soc.,43(5) 2017, 1195–1207.

[3] Abtahi, Mortaza, and Sara Farhangi. Vector-valued spectra of Banach algebra valued continuous func-tions, RACSAM 112.1 (2018): 103-115.

[4] J.Tomiyama,Tensor products of commutative Banach algebras, Tohoku Math. J., 12, (1960) 147–154.

[5] A. Nikou and G. Ofarrell,Banach algebras of vector-valued functions,Glasgow Math J 56 (2014) 419-26.

179

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. ت وپ ول وژی ‐G راس ت ھ ای گ روه ن ی م ، ت وپ ول وژی راس ت ھ ای گ روه ن ی م ی اف ت ه، ت ع م ی م ت وپ ول وژی ک ل ی دی: ک ل م ات۴٢B٣٩ ،۴۵D١٣ م وض وع:

م ق دم ه ١

واژه اس ت. ش ده ای ج اد آن ال ی ز و ھ ن دس ه ت ع اری ف و ق ض ای ا ھ ا، ای ده از ب س ی اری ک ارگ ی ری ب ه از ک ه اس ت ری اض ی ات ع ل م از ق س م ت ت وپ ول وژیگ اوس ش اگ ردان از ی [١] ل ی س ت ی ن ت وس ط ب ار اول ی ن ک ه اس ت آم ده دس ت ب ه م ط ال ع ه و م ان م ع ن ای ب ه ی ون ان ک ل م ه دو ت رک ی ب از ت وپ ول وژی ح ف ظ ھ م ئ وم ورف ی س م ی ع ن ت ب دی الت، از م ع ی ن گ روه وس ی ل ه ب ه ک ه اس ت خ واص ت وپ ول وژی ھ ن دس دی دگ اه از گ ردی د. ١٨۴٧م ع رف س ال در [٢]ب ه ت وان م خ واص ای ن از گ وی ن د م ت وپ ول وژی خ واص آن ب ه ک ه ش ون د م ح ف ظ ھ م ئ وم ورف ی س م ت ح ت ک ی ف خ واص از ب رخ ش ود. م ف ض ای ب رای م خ ت ل ف ت ع اری ف [۴] ھ اس دروف و [٣] ف رش ه م ان ن د م خ ت ل ف ری اض ی دان ان ک رد. اش اره ( (ی پ ارچ ھ م ب ن دی و ف ش ردگ ت ع م ی م ت وپ ول وژی ھ ای گ روه س اخ ت ار ح ال ب ه ت ا ش د. ح اص ل ھ ا س ال از پ س ت ع ری ف ب ه ت ری ن ب رای ھ ا آن ت واف ق ال ب ت ه ک ه دادن د ارائ ه ت وپ ول وژی ن ی م روی ج ب ری خ واص ب رخ و داده ادام ه را زم ی ن ه ای ن در اس اس ن ت ای ب رخ اک ن ون م ا اس ت. گ رف ت ه ق رار م ط ال ع ه م ورد گ س ت رده ط ور ب ه ی اف ت ه

ک ن ی م. م ب ررس را ف ش رده -G ی اف ت ه ت ع م ی م راس ت ت وپ ول وژی ھ ای گ روهرا ت وپ ول وژی خ واص ھ م ان ک ه ش د م ع رف [۵] م رج ١٩٩٧در س ال در [۶] س زار ت وس ط ب ار ن خ س ت ی ن ب رای ھ م ی اف ت ه ت ع م ی م ت وپ ول وژی ن ظ ری هی ن ظ ری ه [٨] م رج ٢٠١٣در س ال در [٧] ح س ی ن م راد ادام ه در دارد. ن ظ ر در ب از ھ ای م ج م وع ه ب رای م ت ن اھ اش ت راک گ رف ت ن ن ظ ر در ب دون

ک رد. ب ی ان را ی اف ت ه ت ع م ی م ت وپ ول وژی ھ ای گ روه

ی اف ت ه ت ع م ی م ت وپ ول وژي م ع رف ٢

ت وپ ول وژي و ھ ن دس ه در ف ع ال ت ح ق ی ق ات ھ اي زم ی ن ه از ی اک ن ون ھ م و ش د ١٩٩٧م ع رف س ال در س زار ت وس ط ب ار ن خ س ت ی ن ی اف ت ه ت ع م ی م ت وپ ول وژیت ع ری ف ب ه ادام ه در ب اش د م ت ع م ی م ق اب ل ت وپ ول وژي ق ض ای اي از ب س ی اری ش رای ط ای ن ح ذف ب ا ک ه دادن د ن ش ان زم ی ن ه ای ن در پ ژوھ ش ران و اس ت

پ ردازی م. م ی اف ت ه ت ع م ی م ت وپ ول وژي دق ی ق

ی اف ت ه ت ع م ی م ت وپ ول وژي ی را P (x) ی ع ن X ت وان ي م ج م وع ه از µ ي زی رم ج م وع ه ب اش د. ن ات ه م ج م وع ه ی X ک ن ی د ف رض .٢ . ١ ت ع ری فف ض اي ی را µ ي ی اف ت ه ت ع م ی م ت وپ ول وژي ھ م راه ب ه X ي م ج م وع ه ب اش د. ب س ت ه دل خ واه اج ت م اع ت ح ت و ب وده ϕ ش ام ل اگ ر ن ام ی م، م µ روي

ن ام ی م. م ت وپ ول وژی) ‐ G ، (GT ی اف ت ه ت ع م ی م ف ض اي ی اخ ت ص ار ب ه ی ا ی اف ت ه ت ع م ی م ت وپ ول وژی ي م ج م وع ه ی ت رت ی ب ھ م ی ن ب ه ش ون د. م ن ام ی ده ب از ‐G ھ ای م ج م وع ه G اع ض اي آن اه ب اش د X روي ی اف ت ه ت ع م ی م ت وپ ول وژي ی G اگ ر

ش ود. م ت ع ری ف ب از -G ي م ج م وع ه م ت م م ص ورت ب ه ب س ت ه ‐G ی م ج م وع ه ی ا ی اف ت ه ت ع م ی م ب س ت ه

اگ ر ش ود م خ وان ده پ ی وس ت ه −G ، X روي f : X → Y ن اش ت ی آن اه ب اش ن د؛ ت وپ ول وژی −G ف ض ای دو Y و X اگ ر .٢ . ٢ ت ع ری فب اش د. ب از −G ، X در f−١(O) ، Y در O ب از -G م ج م وع ه ھ ر ب راي

١٨٠

ف ش رده ‐G راس ت ت وپ ول وژی ‐G ھ ای گ روه ن ی م روی ج ب ری خ واص ب رخ

پ ی وس ت ه −G ھ ای ن اش ت f−١ ھ م و f ھ م اگ ر گ وی ی م Y ب ه X از (ھ م ئ وم ورف ی س م) ھ م س ان ری خ ت −G را f دوس وی ن اش ت .٢ . ٣ ت ع ری فدھ ی م. م ن ش ان X∼=GY ن م اد ب ا و گ وی ی م م ھ م س ان ری خ ت را دو آن آن اه ب اش د داش ت ه وج ود Y و X ب ی ن ھ م س ان ری خ ت −G ی اگ ر ب اش ن د.

م ش م ول ب از ‐G م ج م وع ه ب زرگ ت ری ن را X م ج م وع ه ب ه ن س ب ت A G−درون ، A ⊆ X و ت وپ ول وژی ف ض اي (X,G)ی ھ رگ اه .۴ . ٢ ت ع ری فدھ ی م: م ن م ای ش intG(A) ب ا و ک ن ی م م ت ع ری ف A در

intG(A) = ∪V ⊆ X : V ∈ G,V ⊆ A.intG(A) = A ب ن اب رای ن و اس ت ب از A ، A ∈ G ک ه A ⊆ X م ج م وع ه ھ ر ب راي و X روي G ت وپ ول وژی ف ض اي ھ ر ب راي

دھ ی م. م ن ش ان clG(A) ب ا و ک ن ی م م ت ع ری ف A ش ام ل ب س ت ه −G ھ اي م ج م وع ه ھ م ه اش ت راک را A م ج م وع ه ب س ت ار .۵ . ٢ ت ع ری ف

م ت ن اھ زی رپ وش ش ، X ب از G‐ پ وش ش ھ ر ھ رگ اه ش ود، م ن ام ی ده ف ش رده ‐G ، (X,GX) ی ی اف ت ه ت ع م ی م ت وپ ول وژی ف ض ای .۶ . ٢ ت ع ری فب اش د. داش ت ه

ی ب س ت ه −G ھ ای م ج م وع ه زی ر از خ ان واده ھ ر اگ ر ف ق ط و اگ ر اس ت ف ش رده ‐G ، (X,GX) ی ی اف ت ه ت ع م ی م ت وپ ول وژی ف ض ای .٢ . ٧ گ زارهب اش د. داش ت ه ن ات ه اش ت راک ، م ت ن اھ اش ت راک خ اص ی ت ب ا X

ش ود م راج ع ه [٨] م رج ب ه اث ب ات ب رای اث ب ات.

ب ط وری ه اس ت S در ت وپ ول وژی ی و S گ روه ن ی م ی ش ام ل ی اف ت ه ن ع م ی م (چ پ) راس ت ت وپ ول وژی گ روه ن ی م ی .١ .٢ . ٨ ت ع ری فب اش د. خ ودش ب ه S ف ض ای از پ ی وس ت ه −G ن اش ت ρa (λa چ پ راس ت(ان ت ق ال ان ت ق ال ع م ل a ∈ S ھ ر ب رای

−G ، S در دوت ای ع م ل ک ه ط وری ب ه اس ت τ ی اف ت ه ت ع م ی م ت وپ ول وژی و S گ روه ن ی م ی ش ام ل ی اف ت ه ن ع م ی م ت وپ ول وژی گ روه ن ی م ی .٢ب اش د. پ ی وس ت ه

ی اف ت ه ن ع م ی م ت وپ ول وژی گ روه ن ی م ھ رگ اه ن ام ی م م ی اف ت ه ت ع م ی م ت وپ ول وژی گ روه ی را τ ی اف ت ه ت ع م ی م ت وپ ول وژی ب ا ھ م راه G گ روه .٣ب اش د. پ ی وس ت ه −G آن در م ع وس ع م ل و ب اش د

اص ل اح ام ٣

ب اش د. t ∈ S ھ ر ب رای S ب ه S از پ ی وس ت ه −G ت اب ی ρt : S → St ھ رگ اه اس ت ف ش رده −G ی اف ت ه ت ع م ی م راس ت ت وپ ول وژی گ روه ن ی م Sزی رم ج م وع ه Λ = ∆(S) م ی ن ی م ف رض ھ م چ ن ی ن و اس ت ش ده م ش خ ص E(S) ب ا S از ھ ا خ ودت وان از ای م ج م وع ه

s ∈ S|λS : S → S s.t. G− continuous

ش ود. م S از K = K(S) م ی ن ی م ال آل ای ده از ج ب ری ک ام ل ش رح ب ه م ن ج ر زی ر ن ت ی ج ه ب اش د.

دارد. وج ود خ ودت وان ی S در آن اه ب اش د. ف ش رده −G راس ت ی اف ت ه ت ع م ی م ت وپ ول وژی گ روه ن ی م S ک ن ی د ف رض .٣ . ١ گ زاره

چ ون اس ت ن ات ه I . ب ی ری د ن ظ ر در را ک ن د م ص دق J٢ ⊂ J در ک ه S از J ن ات ه ب س ت ه −G ھ ای م ج م وع ه زی ر ھ م ه از I م ج م وع ه اث ب ات.. s ∈ Hگ ی ری م اس ت. H م ی ن ی م ال ع ض و ح اوی I، S از س ازی ف ش رده −G ب ا پ س ب اش د ش ده م رت ب ش م ول ب ا پ ای ی ن ب ه رو I اگ ر . S ∈ I ک هHs = H ، H از م ی ن ی م ال ی ت ب ا رو ای ن از . HsHs ⊂ HHHs ⊂ Hs و اس ت ف ش رده −G و ن ات ه Hs = ρst|t ∈ H ب ن اب رای ن

ک ه آن ج ا از دارد. وج ود ts = s ب ا t ∈ H پ س اس ت.

t ∈ W = ρ−١s (s) ∩H = r|rs = s ∩H

.s٢ = s و s ∈ W ، W = H رو ای ن از ک ن د. م ص دق W ٢ ⊂ W در و ب س ت ه −G ، ت ه ن ا W

ب اش د: ف ش رده −G راس ت ی اف ت ه ت ع م ی م ت وپ ول وژی گ روه ن ی م S ک ن ی د ف رض .٣ . ٢ ق ض ی ه

ھ س ت ن د ب س ت ه −G ، S م ی ن ی م ال چ پ ھ ای آل ای ده اس ت. م ی ن ی م ال چ پ آل ای ده ش ام ل S چ پ س م ت ای ده ھ ر .١

اس ت. K = K(S) ط رف ه دو آل ای ده ک وچ ت ری ن دارای S .٢

م ع ادل ن د: زی ر اح ام ، e ∈ S خ ودت وان ی ب رای و ھ اس ت خ ودت وان ش ام ل K .٣

181

پ ی رع ل رض ا ، ب ای م ان ن س ت رن

e ∈ K (آ)K = SeS (ب)

اس ت. م ی ن ی م ال چ پ آل ای ده Se (ج)اس ت. م ی ن ی م ال راس ت آل ای ده eS (د)

اس ت. S از م اک س ی م ال زی رگ روه eSe (ه)

از ب رخ ب رای اس ت eS ف رم از م ی ن ی م ال راس ت آل ای ده ھ ر ؛ e ∈ K خ ودت وان از ب رخ ب رای اس ت Se ف رم از م ی ن ی م ال چ پ آل ای ده ھ ر .۴.e ∈ K خ ودت وان

K = ∪eSe|e ∈ E(K) = ∪eS|e ∈ E(K) = ∪Se|e ∈ E(K) .۵

م ی ن ی م ال راس ت آل ای ده ھ م ان ن د م اک س ی م ال ھ ای زی رگ روه ھ س ت ن د؛ ای زوم ورف ی −G ج ب ری ص ورت ب ه K م اک س ی م ال ھ ای گ روه زی ر ھ م ه .۶ھ س ت ن د. ای زوم ورف ی −G ت وپ ول وژی ص ورت ب ه ،

ح اص ل ض رب ب ا E(Se) × eSe × E(eS) ب ا اس ت ای زوم ورف ی −G ج ب ری ص ورت ب ه K ، e ∈ K خ ودت وان ھ ر ب رای .٧−G ، eSe × E(eS) راس ت گ روه ب ا ج ب ری ص ورت ب ه م ی ن ی م ال راس ت آل ای ده و .(u, v, w)(x, y, z) = (u, vwxy, z)

. ھ س ت ن د ای زوم ورف ی −G ، eSe× E(Se) چ پ گ روه ب ا ج ب ری ص ورت ب ه م ی ن ی م ال چ پ آل ای ده و ھ س ت ن د ای زوم ورف ی

از چ ال پ ذی ر ت ع وی ض م ج م وع ه زی ر ی Λ ک ه ط وری ب ه ب اش د ف ش رده −G راس ت ی اف ت ه ت ع م ی م ت وپ ول وژی گ روه ن ی م S ک ن ی د ف رض .٣ . ٣ ق ض ی ه، Kاز م اک س ی م ال ھ ای گ روه زی ر ت م ام و R = K آن اه اس ت. ب س ت ه −G ک ه اس ت R م ی ن ی م ال راس ت آل ای ده دارای S ک ن ی د ف رض و اس ت S

ھ س ت ن د. ای زوم ورف ی −G ت وپ ول وژی ص ورت ب ه و ب س ت ه −Gی K از م اک س ی م ال زی رگ روه ھ ر و R = K پ س . SR ⊂ R = R ک ه دھ د م ن ت ی ج ه ، s ∈ Λھ م ه ب رای sR = Rs ⊂ R چ ون اث ب ات.

ک ن د. م پ ی روی (۴) و (١) ، ٣ . ٢ ق ض ی ه از ب ق ی ه اس ت. م ی ن ی م ال چ پ آل ای ده

S از چ ال پ ذی ر ت ع وی ض زی رم ج م وع ه Λ ک ه ط وری ب ه ب اش د ف ش رده −G راس ت ی اف ت ه ت ع م ی م ت وپ ول وژی گ روه ن ی م S ک ن ی د ف رض .۴ . ٣ ق ض ی هو اگ ر اس ت م ی ن ی م ال راس ت آل ای ده ی K رو، ای ن از اس ت چ ال اس ت آن چ پ آل ای ده ش ام ل ک ه K م اک س ی م ال زی رم ج م وع ه ھ ر آن اه اس ت.

ت وپ ول وژی ل ح اظ از و ب س ت ه −G ، K م اک س ی م ال ھ ای گ روه زی ر ت م ام ح ال ت، ای ن در ؛ ب اش د ب س ت ه −G م اک س ی م ال زی رگ روه دارای K اگ ر ت ن ه ا.ھ س ت ن د ای زوم ورف ی −G

، s ∈ Λ ھ م ه ب رای اس ت st = ts ب ن اب رای ن ب اش د. S از L م ی ن ی م ال چ پ آل ای ده ی در م ش م ول K م اک س ی م ال زی رگ روه ی G گ ی ری م اث ب ات.ش ود. م دن ب ال راح ت ب ه م ان ده ب اق اظ ه ارات ب اش د. L ب ا ب راب ر ب ای د ک ه ، t ∈ S

م راج

[1] A.V.Arhangel ski,M.Thachenko.Topological groups and structures. Atlantis Studies in Mathmathics, Vol, 1 , At-lantis Press/ world Scientific, Amsterdam-paris,2008

[2] E.Bohn,j.Lee. Semi-topological groups.Amer.Math Monthy.72(1965), 996-998.

[3] M.S.Bosan, M.D.Khan and Ljubisa, D.R.Kocinac. On s-topological groups. Math-ematica. Moravica, Vol. 18-2(2014), 35-44.

[4] P.Bhattacharyy. and Lahiri, B.K. Semi-generalized closed sets in topology. Math, 29(3)(1987), 375-382

[5] A.Csaszar. Generalized open sets. Acta Math. Hungar., 75(1997), 65-87

[6] M.Caldas. On g-closed sets and g-continuose mappings. Kynung Pook Math.Jr., 33(2).

[7] M.Caldas. Semi T 12spaces. Atas Sem. Bras. Ahal., 40(1994),191-196.

27(4), (2013) 567–575.

[8] M. Hussain, M. Khan and C. Ozel, More on generalized topological groups, Creat, Math. Inf. 22, (2013), 47–51.

[9] J. R Berglund, H. D. Junghenn, P. Milnes. Compact Right Topological Semigroups and Generalizations of AlmostPeriodicity.

182

م وج ھ ای ق اب ک ان ون دوگ انج و* ح ق ی ق ت اح س ان

ای ران) س ب زوار، س ب زواری، ح ی م (دان ش اه

وج ود ف رک ان س دام ن ه در ف ش رده گ اه ت ی ه و زم ان ی دام ن ه در س ری ک اھ ش ب ا ψ ق اب م وج ی ک ه دھ ی م م ن ش ان م ق ال ه ای ن در چ ی ده:ت ع داد دی ر ط رف از ش ود، ت ول ی د م ول دھ ا از دل خ واھ ت ع داد ت وس ط ت وان د ن م ق اب ک ان ون دوگ ان ک ه ک ن د م ت ول ی د م وج س ی س ت م و دارد

ش ود. م ت ول ی د ت اب ی ت وس ط ψ از ک ان ون غ ی ر دوگ ان ن ام ت ن اھ ی دھ د ت ش ی ل واح د اف راز ی آن ف وری ه ت ب دی ل ھ ای ات س اع ک ه ψ م ح دود ت اب ھ ر و ک وچ ک اف ق در ب ه ان ت ق ال پ ارام ت رھ ای ب رای پ ای ان در

ب ا ψ ھ ای ات س اع از م ت ن اھ خ ط ت رک ی ب ی ت وس ط ق اب دوگ ان ای ن دارد. م وج س اخ ت ار ن ی ز آن دوگ ان ک ه ک ن د م ت ول ی د م وج ق ابش ود. م ت ول ی د م ع ل وم ض رای ب

غ ی رک ان ون دوگ ان ، م وج ق اب ک ان ون دوگ ان ، م وج ق اب ک ل ی دی: ک ل م ات۴٢B٣٩ ،۴۵D١٣ م وض وع:

م ق دم ه ١

داری م. (Rd)L٢ن ی از در زی ر ع م ل ر س ه ب ه اب ت دا در

ب اش د. d× d ن ام ن ف رد م ات ری س ی A و m ∈ Z و k ∈ Zd ک ن ی م ف رض .١ . ١ ت ع ری فک ن ی م م ت ع ری ف ص ورت ای ن در

Tk : L٢(Rd) → L٢(Rd), Tkf(x) = f(x− k) (١ ان ت ق ال (ع م ل رDA : L٢(Rd) → L٢(Rd), DA = | detA|١/٢f(Ax) (٢ ات س اع (ع م ل ر

Em : L٢(Rd) → L٢(Rd), Emf(x) = e٢πim.xf(x) (٣ م دوالس ی ون (ع م ل ر

ق اب از رده ۴ی م وج ھ ای ق اب دھ ی م. م ن ش ان D ب ا را DA اخ ت ص ار ب ه و ن ام ی م ات س اع م ات ری س A را ف وق ت ع ری ف در A م ات ری سف ض ای در م وج ھ ای ق اب ی ادآوری ض م ن درادام ه ان د، داده ن ش ان را خ ود ک ارای م ه ن دس و ع ل وم ھ ای ش اخ ه از ب س ی اری در ک ه ھ س ت ن د ھ ا

پ رداخ ت. خ واھ ی م ھ ای ق اب چ ن ی ن دوگ ان ب ررس ب ه L٢(Rd) ھ ی ل ب رتی خ ان واده از اس ت ع ب ارت Ψ ت وس ط ت ول ی دش ده م وج س ی س ت م ی ،r ∈ N ک ه Ψ := ψiri=١ ⊂ L٢(Rd) ک ن ی د ف رض

Dj Tkψi; j ∈ Z, k ∈ Zd, i = ١, ..., r (١)

ب ه را ب اش د آن ب رای ی ه) م ت ع ام د پ ای ه ی ا ری س پ ای ه ی ا (ق اب ب س ل ی ف وق دن ب ال ه ھ رگ اه اس ت. ان ت ق ال ع م ل ر Tk و ات س اع ع م ل ر Dj ک هش ود. م ن ام ی ده ψiri=١ ت وس ط ت ول ی دش ده ی ه) م ت ع ام د پ ای ه ی ا م وج ری س پ ای ه ی ا م وج (ق اب م وج ب س ل دن ب ال ه ی ت رت ی ب

ن ام ن د. م م ت ع ام د) م وج ی ا ری س م وج ق اب، ب س ل(م وج رام وج ψiم ول دھ ای م ج م وع ه ح االت درای نھ س ت ن د. م وج ھ ای ق اب ش ن اس ای ب ا ارت ب اط در زی ر ق ض ای ای

١Translation٢Dilation٣Modulation۴Wavelet frames

١٨٣

ج و ح ق ی ق ت اح س ان

ک ن ی د ف رض ن ی ز و ψ ∈ L٢(R) و b > ٠ ، a > ١ ک ن ی د ف رض .١ . ٢ ق ض ی ه

B :=١b

sup|ξ|∈[١,a]

∑

j,k∈Z

|ψ(ajξ)ψ(ajξ +k

b)| < ∞.

اگ ر ع الوه ب ه اس ت. B ک ران ب ا ب س ل دن ب ال ه ی DjaTbkf م وج س ی س ت م آن اه

A :=١b

inf|ξ|∈[١,a]

(∑

j∈Z|ψ(ajξ)|٢ −

∑

k =٠

∑

j∈Zψ(ajξ)ψ(ajξ +

k

b)|) > ٠,

اس ت. B و A ھ ای ک ران ب ا L٢(R) ب رای ق اب ی DjaTbkψ آن اه ب اش د

ب ب ی ن ی د. را [١] م رج از ١١٬٢٬٣ ق ض ی ه اث ب ات ب رای اث ب ات.

از DjaTbkψj,k∈Z و Dj

aTbkψj,k∈Z م وج س ی س ت م دو ک ن ی د ف رض .ψ, ψ ∈ L٢(R) و b > ٠ ،a > ١ ک ن ی م ف رض .١ . ٣ ق ض ی هک ن ن د: ص دق زی ر ش رای ط در اگ ر ھ س ت ن د ق اب دوگ ان Dj

aTbkψj,k∈Z و DjaTbkψj,k∈Z آن اه ب اش ن د ب س ل ھ ای خ ان واده

∑

j∈Zψ(ajξ)

ˆψ(ajξ) = b a.eξ ∈ R, (٢)

ˆψ(ξ)ψ(ξ + q) = ٠ a.eξ ∈ R,٠ = q ∈ b−١Z. (٣)

ب ب ی ن ی د. [٢] م رج ٢ ق ض ی ه در را اث ب ات اث ب ات.

م وج س اخ ت ار ب دون ق اب ک ان ون دوگ ان ٢

ک ن ی م. م ارائ ه را ن دارد م وج س اخ ت ار آن ک ان ون دوگ ان ک ه م وج ق اب ی از م ث ال اک ن ون

ک اف ق در ب ه ϵ > ٠ ب رای ب اش د. ψj,kj,k∈Z م وج ی ه م ت ع ام د پ ای ه ی م ول د ψ ک ن ی د ف رض .٢ . ١ م ث الواض ک ن د، م ت ول ی د م وج ری س پ ای ه ی ک ه ک ن ی م م ت ع ری ف η := ψ + ϵDψ ص ورت ب ه را η ک وچ ی ب ن اب رای ن اس ت، ψj,k ی ه م ت ع ام د پ ای ه از دن ب ال ه زی ر ی DjTkDψ = Dj+١T٢kψ دن ب ال ه ک ه اس ت

ک ران ب ا ب س ل )دن ب ال ه١ ± ϵ١/٢(٢ ھ ای ک ران ب ا ری س پ ای ه ی ηj,k دن ب ال ه ϵ ∈

(٠,١

)ھ ر ب رای ل م ب ن اب ر ن ت ی ج ه در اس ت. B ≤ ١

اس ت زی ر ص ورت ب ه و ش ده اس ت ف اده S−١η ηj,k م ح اس ب ه ب رای [٣] در ن ت ی ج ه ای ن از ص ری ح ا اس ت.

S−١η ηj,k =

∑∞n=٠(−ϵ)nψj−n,٠, ∀j ∈ Z, k = ٠

ψj,k, ∀j ∈ Z, k ∈ ٢Z + ١(۴)

اس ت. ψj,kj,k∈Z ت واب از م ت ن اھ خ ط ت رک ی ب ص ورت ب ه k = ٠ ب رای ن ی ز وب ا .S−١

η ηj,٠ = ϕj,٠ ک ه ط وری ب ه ن دارد وج ود ϕ ∈ L٢(R) واق در ن دارد م وج س اخ ت ار ق اب دوگ ان ک ه دھ ی م م ن ش ان k = ٠ ح ال ت دری ع ن ک ن د ت ول ی د ق اب دوگ ان ی ک ه ب اش د م وج ود ϕ ∈ L٢(R) ی اگ ر خ ل ف ف رض

ϕj,k = S−١η ηj,k, ∀j, k ∈ Z

.ϕ = ψ ب ن اب رای ن و T١ψ = T١ϕ ی ا ψ٠,١ = ϕ٠,١ ک ه ش ود م ن ت ی ج ه k = ١ و j = ٠ ب ا (۴) از آن اهداری م j = ٠ ب ا (۴) در

ψ = ϕ٠,٠ =∞∑

n=٠(−ϵ)nψ−n,٠ = ψ +

∞∑

n=١(−ϵ)nψ−n,٠,

پ س∞∑

n=٠(−ϵ)nψ−n,٠ = ٠ a.e.

ت اب ی ت وس ط ت ن ه ا ت وان د ن م ηj,k ق اب دوگ ان ن ت ی ج ه در اس ت. ت ن اق ض در ψj,kj,k∈Z ی ه م ت ع ام د پ ای ه ب ودن ω−م س ت ق ل ب ا ای ن ک هش ود. ت ول ی د

184

م وج ھ ای ق اب ک ان ون دوگ ان

ب ب ی ن ی د. [۴] در ت وان ی د م را آن اث ب ات اس ت ف وق م ث ال از ت ری ک ل ح ال ت زی ر ق ض ی ه

ک ه ط وری ب ه دارد وج ود ψ ∈ L٢(R) ق اب م وج ی .٢ . ٢ ق ض ی ه

دارد. ف ش رده گ اه ت ی ه و اس ت C∞ ت اب ع ψ .١

ن ی س ت. ت اب ی ت وس ط ش ده ت ول ی د م وج س ی س ت م ی ص ورت ب ه ψ ت وس ط ش ده ت ول ی د ق اب ک ان ون دوگ ان .٢

ان د. م وج ھ ای ق اب دوگ ان ج ف ت ی ψ و ψ ک ه ط وری ب ه دارد وج ود ψ ن ام ت ن اھ ت ع داد .٣

ک ن ی م. م ن ظ ر ص رف ب ودن ط والن دل ی ل ب ه آن اث ب ات از و ب ی ان ف ق ط را اس ت ق ب ل ق ض ی ه از ت ع م ی م ک ه زی ر ق ض ی ه ادام ه در

ک ه ط وری ب ه دارد وج ود ψ ∈ L٢(R) ق اب م وج ی J ∈ N ھ ر ب رای .٢ . ٣ ق ض ی ه

دارد. ف ش رده گ اه ت ی ه و اس ت C∞ ت اب ع ψ .١

ن ی س ت. ت اب ٢J از ک م ت ر ب ا ش ده ت ول ی د م وج س ی س ت م ی ص ورت ب ه ψ ت وس ط ش ده ت ول ی د ق اب ک ان ون دوگ ان .٢

ان د. م وج ھ ای ق اب دوگ ان ج ف ت ی ψ و ψ ک ه ط وری ب ه دارد وج ود ψ ن ام ت ن اھ ت ع داد .٣

م وج س اخ ت ار ب ا ک ان ون غ ی ر دوگ ان ص ری س اخ ت ن ٣

گ رف ت. ن ظ ر در م وج ق اب ی م ول د ع ن وان ب ه ت وان م را ب دھ د واح د اف راز ی ت ش ی ل آن ف وری ه ت ب دی ل ھ ای ات س اع ک ه ک ران دار ت اب ھ ر

ب ا م ق دار ح ق ی ق ت اب ی ψ ∈ L∞(R) ک ن ی د ف رض ن ی ز و ψ ∈ L٢(R) و a > ١ ، n ∈ N ک ن ی د ف رض .٣ . ١ ل مو c ∈ Z ک ه ب اش د suppψ ⊂ [−ac,−ac−n] ∪ [ac−n, ac]

∑

j∈Zψ(ajξ) = ١ a.e.ξ ∈ R. (۵)

ص ورت ب ه ش ده ت ع ری ف ϕ ت اب و ψ ت اب آن اه .b ∈ (١−٠,٢a−c] ک ن ی د ف رض ھ م چ ن ی ن

ϕ(x) = bψ(x) + ٢bn−١∑

j=١a−jψ(a−jx), x ∈ R, (۶)

ک ن ن د. م ت ول ی د L٢(R) ب رای DjaTbkϕj,k∈Z و Dj

aTbkψj,k∈Z ھ ای ق اب دوگ ان

ف وری ه ت ب دی ل ب ودن خ ط و (۶) در ϕ ت ع ری ف ب ه ت وج ه ب ا ھ م چ ن ی ن اس ت، R \ ٠ در ف ش رده گ اه ت ی ه دارای ψ ک ه داری م ت وج ه اب ت دا اث ب ات.داری م

ϕ(ξ) = bψ(ξ) + ٢bn−١∑

j=١ψ(ajξ).

k ب رای زی را ک ن ن د، م ت ول ی د م وج ب س ل ھ ای دن ب ال ه ϕ و ψ ت واب (١٬٢) ب ن اب رق ض ی ه ازط رف اس ت. ف ش رده گ اه ت ی ه دارای ن ی ز ϕ درن ت ی ج هک ه ط وری ب ه اس ت م وج ود ای M > ٠ ب زرگ ک اف ق در ب ه ھ ای

sup|ξ|∈[١,a]

∑

j,k∈Z

|ψ(ajξ)ψ(ajξ +k

b)| ≤ M

∑

j

|ψ(ajξ)| = M < ∞.

م ش اب ه اsup

|ξ|∈[١,a]

∑

j,k∈Z

|ϕ(ajξ)ϕ(ajξ +k

b)| < ∞.

ق ض ی ه ش رای ط دھ ی م ن ش ان اس ت ک اف م ن ظ ور ای ن ب رای ک ن ن د. م ت ول ی د م وج ھ ای ق اب دوگ ان ϕ و ψ دھ ی م م ن ش ان اث ب ات ات م ام ب رایک ن ی م. ب ررس [١, a] و [−a,−١] ھ ای ب ازه روی را ش رط ای ن اس ت ک اف (۵) ش رط در aj−ات س اع ت ن اوب ب ه ت وج ه ب ا اس ت. ب رق رار (١٬٣)

185


ج م الت خ اص ح ال ت در اس ت. ن اص ف ر (٢) س ری در ج م ل ه م ت ن اھ ت ع داد ف ق ط ب ازه دو ای ن روی اس ت، ف ش رده گ اه ت ی ه دارای ψ چ ونان د. ن اص ف ر j = c− n, c− n+ ١, ..., c− ١

اس ت: زی ر ص ورت ب ه ψ ھ ای ات س اع گ اه ت ی ه

suppψ(ac−n.) ⊂ [−an,−١] ∪ [١, an],

suppψ(ac−n+١.) ⊂ [−an−١,−١a

] ∪ [١a, an−١],

ب االخ ره وsuppψ(ac−١.) ⊂ [−a,−a−n+١] ∪ [a−n+١, a].

داری م |ξ| ∈ [١, a] ھ ر ب رای ف رض ب ن اب ر

١ =

(∑

j∈Zψ(ajξ)

)٢

=

(c−١∑

j=c−nψ(ajξ)

)٢

=[ψ(ac−nξ) + ψ(ac−n+١ξ) + ...+ ψ(ac−١ξ)

]٢

= ψ(ac−nξ)[ψ(ac−nξ) + ٢ψ(ac−n+١ξ) + ...+ ٢ψ(ac−١ξ)

]

+ψ(ac−n+١ξ)[ψ(ac−n+١ξ) + ٢ψ(ac−n+٢ξ) + ...+ ٢ψ(ac−١ξ)

]

+...+ ψ(ac−١ξ)[ψ(ac−١ξ)

]

=١b

c−١∑

j=c−nψ(ajξ)ϕ(ajξ) =

١b

∑

j∈Zψ(ajξ)ϕ(ajξ) .

ک ن ن د. م ص دق (٢) ش رط در ϕ و ψ ب ن اب رای نو suppψ(.± q) ⊂ B(±q, ac) آن اه ب اش د r ش ع اع و x م رک ز ب ه ب س ت ه گ وی ن م ای ش B(x, r) اگ ر ط رف از، داری م |q| ≥ ٢ac ک ه ھ ن ام درن ت ی ج ه .suppϕ ⊂ [−ac,−ac−٢n+١] ∪ [ac−٢n+١, ac] ⊂ B(٠, ac)

min |b−١Z \ ٠| = ١b ≥ ٢ac چ ون b ≤ ١−٢a−c ان ت خ اب ب ا رو ازای ن . suppψ ∩ suppϕ = ∅

اث ب ات و ش ود م م ح ق ق (١٬٣) ق ض ی ه ش رای ط ب ن اب رای ن ک ن ن د. م ص دق (٣) ش رط در q ∈ b−١Z \ ٠ ھ ر ب رای ت اب ھ ردو ک ه ش ود م ن ت ی ج هاس ت. ت م ام ل م

م وارد در را آن ک ان ون غ ی ر دوگ ان و م وج ق اب اھ م ی ت ب اش ن د. م ص ری ف رم ب ه ک ان ون غ ی ر دوگ ان دارای ھ ا ق اب چ ن ی ن دھ ی م م ن ش انک ن ی م. م خ الص ه زی ر

. ک ان ون دوگ ان ح ال ت ب رخ الف ان د م رت ب ه ھ م ھ م وار ای چ ن دج م ل ه ی ع ن دارن د ای م ش اب ه ف رم ϕ و ψ : م ش اب ه و ۵ ص ری ف رم .١

دارن د. ۶ ف وری ه دام ن ه در ف ش رده گ اه ت ی ه ϕ ھ م و ψ ھ م .٢

ψ(x) = O(x−r) ی ع ن ک ن د م ص دق lim|x|→∞ xrψ(x) = ٠ در ψ م ول د ψ ∈ Cr٠(R) ب رای زم ان: دام ن ه در ٧ س ری ک اھ ن ده .٣دارد. را وی ژگ ھ م ی ن ن ی ز ϕ دوگ ان م ول د .|x| → ∞ وق ت

دارد r ∈ N ∪ ٠ م رت ب ه از ش ده ت ب اھ ی ده گ ش ت اور ψ م ول د ψ ∈ Cr٠(R) ب رای ک ل ح ال ت ٨:در ش ده ت ب اھ ی ده گ ش ت اور م رت ب ه ب زرگ ت ری ن .۴زی را

٠ =dmψ

dξm(٠) = (−٢πi)m

∫

Rxmψ(x)dx, m = ٠, ..., r.

دارد. را وی ژگ ھ م ی ن ن ی ز ϕ دوگ ان م ول د۵Explicit۶Fourier domain٧Fast decay٨Vanishing moments

186

م وج ھ ای ق اب ک ان ون دوگ ان

.ϕ و ψ ھ م چ ن ی ن ھ س ت ن د، زوج و ح ق ی ق ت واب ϕ و ψ: ت ق ارن .۵∑

j∈Z g(٢jx) = ١ a.e.x ∈ R در ک ه س ازی م م ط وری را g ∈ L٢(R) ت اب ب ع د م ث ال دردس ت ب ه ن ی ز a ب ا ٢ ت ع وی ض ب ا a > ١ ح ق ی ق ات س اع ھ ر ب رای ن ت ی ج ه ای ن ک ه اس ت ب دی ه اس ت. a = ٢ ات س اع ھ ر ب رای م ت ن اظ ر ک ه ک ن د ص دق

آی د. م

و f(٢m − δ) = ٠ در ک ه [٢m − δ,٢m + δ] روی f ک ران دار ت اب ھ ر و ٠ < δ ≤ ٢m/٣ ھ ر و m ∈ Z ھ ر ب رای .٣ . ٢ م ث الک ن ی م م ت ع ری ف زی ر ص ورت ب ه را h١ ک ن د، م ص دق f(٢m + δ) = ١

h١(x) =

f(x) x ∈ B(٢m, δ),١ x ∈

(٢m + δ,٢m+١ − ٢δ

),

١ − f(x/٢) x ∈ B(٢m+١,٢δ),٠ ای ن ص ورت غ ی ر در

(٧)

ع الوه ب ه ب اش د. پ ی وس ت ه f اگ ر اس ت پ ی وس ت ه h١ ∈ L٢(R) ت اب

∑

j∈Zh١(٢jx) =

١ x > ٠,٠ x ≤ ٠.

داری م h٢ ∈ L٢(R) ب رای روش ھ م ی ن ب ردن ک ار ب ه ب ا و

∑

j∈Zh٢(٢jx) =

٠ x ≥ ٠,١ x < ٠.

در ک ه ب اش د ای چ ن دج م ل ه f ک ن ی م ف رض اگ ر ھ م چ ن ی ن ک ن ی م. ت ع ری ف g = h١ + h٢ ص ورت ب ه را g اس ت ک اف ب ن اب رای نرس ی د. ھ م واری g ب ه ت وان م رون د ای ن ادام ه ب ا . g ∈ C١(R) آن اه ک ن د ص دق f ′(٢m + δ) = f ′(٢m − δ) = ٠

(درواق b = ١ ان ت ق ال پ ارام ت ر و ٢ ات س اع ب ا م وج ھ ای ق اب دوگ ان و ک ن ی م م ط راح را ق ب ل م ث ال ھ ای روش ب ع د م ث ال درآوری م. م دس ت ب ه را ( گ ی ری م م ١ را b ک ار راح ت ب رای ام ا ش ود گ رف ت ه ن ظ ر در ت وان د م b ∈

(٠,١

]ھ ر

ت وان د م f م ث ال ب رای اس ت، f(١/٢) = ٠ و f(١/۴) = ١ در ص ادق و[١/۴,١/٢

]ب ازه روی پ ی وس ت ه ت اب ی f ک ن ی م ف رض .٣ . ٣ م ث ال

ب اش د: زی ر ص ورت ب ه ت واب

f(x) = ٢ − ۴x, (٨)f(x) = ٢)٨۴x٢ − ٨x+ ٢)(١x− ٢(١, (٩)f(x) = −١۶(٣٢٠x٣ − ١٩٢x٢ + ۴٢x− ٢)(٣x− ٣(١, (١٠)f(x) = ٣٢(۴۴٨٠x۴ − ٣٨۴٠x٣ + ١٢٨٠x٢ − ١٩٢x+ ٢)(١١x− ١)۴, (١١)

f(x) =١٢

+١٢

cosπ(۴x− ١). (١٢)

در ت رت ی ب ب ه (١١) و (١٠) ت واب ن ق اط ھ م ی ن در ن ی ز و ک رده ص دق f ′(١/۴) = f ′(١/٢) = ٠ در (١٢) ت ا (٨) در ش ده م ع رف f ت واب اس ت. ص ف ر ھ م س وم م رت ب ه م ش ت ق و دوم م رت ب ه م ش ت ق

ک ن ی م م ت ع ری ف زی ر ص ورت ب ه را ψ ∈ L٢(R) ق ب ل م ث ال م ش اب ه

ψ(ξ) =

١ − f(٢|ξ|) |ξ| ∈[١/٨,١/۴

],

f(|ξ|) |ξ| ∈(١/۴,١/٢

],

٠ ای ن ص ورت غ ی ر در(١٣)

ξ ∈ R ھ ر و i ∈ Z ھ ر ب رای زی را ک ن د م ص دق (۵) در suppψ ⊂[

− ١/٨−,١/٢]

∪[١/٨,١/٢

]ب ا ψ درن ت ی ج ه

پ س ٢i|ξ| ∈[١/٨,١/۴

]ی ا ٢i|ξ| /∈

[١/٨,١/۴

]

∑

j

ψ(٢jξ) = ٠ + ٠ + ...+ ١ − f(٢|ξ|) + f(٢|ξ|) + ٠ + ...+ ٠ = ١

187


راب ط ه ط ب ق ل م ای ن پ ی رو در ب رد، ک ار ب ه b = ١ و n = ٢ ، c = −١ ب ا را (٣٬١) ل م ت وان م رو ازای ناس ت زی ر ص ورت ب ه ϕ ∈ L٢(R) دوگ ان م ول د ϕ(ξ) = bψ(ξ) + ٢b

∑n−١j=١ ψ(ajξ)

ϕ(ξ) =

٢[١ − f(۴|ξ|)

]|ξ| ∈

[١/١۶,١/٨

],

١ + f(٢|ξ|) |ξ| ∈(١/٨,١/۴

],

f(|ξ|) |ξ| ∈(١/۴,١/٢

],

٠ ای ن ص ورت غ ی ر در

(١۴)

ک ن ن د. م ت ول ی د L٢(R) ب رای Dj٢Tkϕj,k∈Z و Dj

٢Tkψj,k∈Z ھ ای ق اب دوگ ان ϕ و ψ پ سش ود. م ت ول ی د ت اب ی ب ا ف ق ط م وج ق اب ھ ر و اس ت ش ده داده ق رار b = ١ ھ ا م وج ای ن در ان ت ق ال پ ارام ت ر

،r = ٠,١,٢,٣ ب ا ت رت ی ب ب ه آن اه ب اش د (١٢) ت ا (٨) ھ ای ت س اوی ص ورت ب ه ش ده ت ع ری ف ای چ ن دج م ل ه ی ψ ∈ L٢(R) ک ن ی د ف رضsuppψ ⊂

[− ١/٨−,١/٢

]∪[١/٨,١/٢

]ب ا ھ م وار ای چ ن دج م ل ه ϕ و ψ و ان د زوج و ح ق ی ق ϕ و ψ م ول دھ ای ع الوه ب ه .ψ ∈ Cr(R)

س ادگ ب ه ϕ ب ن اب رای ن و ψ ص ری ف رم ب ب ی ن ی د). را ٢ و ١ ن م ودارھ ای ان د( ف ش رده ک ه suppϕ ⊂[− ١/١−,١/٢۶

]∪[١/١۶,١/٢

]و

sin(٢παx)/(πx)n ف رم ب ه α ∈ Q و n ≥ ٢ + r ع ددص ح ی ب رای و ان د cos و sin از م ت ن اھ ت رک ی ب آن ه ا ک ل ح ال ت در ش ون د، م ی اف تھ س ت ن د. cos(٢παx)/(πx)n و

ک ه ٠ ≤ f(x) ≤ ١ ھ ر ب رای (ح ت ش د ت ع ری ف ق ب ل در ک ه ھ ای f ھ م ه ب رای ک ه ک ن ی م م ادع ا اک ن ونDj

٢Tkϕ ق اب دوگ ان ھ ای ک ران C٢ = ۵ و C١ = ٧٢ و Dj

٢Tkψ ق اب ھ ای ک ران C٢ = ١ و C١ = ١٢ ،( x ∈

[١/۴,١/٢

]

داری م (٢jξ + k) /∈[١/۴,١/٢

]ای k ب رای ای ن ه و suppψ ب ه ت وج ه ب ا ادع ا اث ب ات ب رای ھ س ت ن د).

∑

k =٠

∑

j∈Z|ψ(٢jξ)ψ(٢jξ + k)| = ٠, ξ ∈ R.

ش ون د م ب رآورد زی ر ص ورت ب ه ق اب ھ ای ک ران (١٬٢) ق ض ی ه ب ن اب ر ط رف از

C١ = inf|ξ|∈[

١/۴,١/٢]∑

j∈Z|ψ(٢jξ)|٢, C٢ = sup

|ξ|∈[

١/۴,١/٢]∑

j∈Z|ψ(٢jξ)|٢.

داری م |ξ| ∈[١/۴,١/٢

]ب رای (١٣) ت ع ری ف ب ن اب ر ھ س ت ن د. Dj

٢Tkψ ق اب پ ای ی ن و ب اال ک ران ه ای ت رت ی ب ب ه C١ و C٢ ک ه∑

j∈Z |ψ(٢jξ)|٢ = f(|ξ|)٢ +(١ − f(|ξ|)

)٢= ١ − ٢f(|ξ|) + ٢f(|xi|)٢,

C٢ = maxx∈[α,β] ١ − ٢x+ ٢x٢ و C١ = minx∈[α,β] ١ − ٢x+ ٢x٢ = ١٢ ب اف رض ب ن اب رای ن و

وک ه ش ود م ن ت ی ج ه ٠ ≤ f(x) ≤ ١ ، x ∈

[١/۴,١/٢

]ھ ر ب رای چ ون ، β := max١/۴≤x≤١/٢ f(x) و α := min١/۴≤x≤١/٢ f(x)

م ث اب ت را ادع ا ای ن ک ه C٢ = ۵ و C١ = ٧٢ ک ه دی د ت وان م ن ی ز ق اب دوگ ان ب رای م ش اب ه ط ور ب ه .C٢ = ١ ب ن اب رای ن و β = ١ و α = ٠

ک ن د.

م راج

[1] O. Christensen, An Introduction to Frames and Riesz Bases. Birkhäuser Boston, 2003.

[2] C. Chui, X. Shi, Orthonormal wavelets and tight frames with arbitrary real dilations, Appl. Comput. Harmon. Anal. 9(2000), 243-264.

[3] C. Chui, X. Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAMJ. Math. Anal. 24 (1993), 263-277.

[4] I. Daubechies, B. Han, The canonical dual frame of a wavelet frame, Appl. Comput.Harmon. Anal. 12 (2002 ), 269-285. I. Du

188

م ق ادی ر م ات ری س ھ س ت ه ھ ای ب ا ھ م وردا ت واب ش م س* م ه دی

ای ران) ک اش ان، ک اش ان، (دان ش اه

م ق دار ب ردار ھ م وردای ت واب ب رای ک ه م ش ون د م ع رف ھ س ت ن د ھ س ت ه از ت ع م ی م ی ک ه م ق دار م ات ری س ھ س ت ه ھ ای رده م ق ال ه ای ن در چ ی ده:م گ ی رن د. ق رار اس ت ف اده م ورد

ھ ی ل ب رت. ف ض ای م ق دار، م ات ری س ھ س ت ه زی ان، ت اب ت پ ی م ان ه ای، گ روه ھ م وردا، ت واب گ روه، ع م ل ک ل ی دی: ک ل م ات١١H۵۴ م وض وع:

م ق دم ه ١

م ش ود. ت وص ی ه [۵ ،۴] م راج خ وان دن ش اخ ه ای ن ک ارب ردھ ای م ش اھ ده ع ن وان ب ه دارن د. م اش ی ن ی ادگ ی ری در ف راوان ک ارب رد ھ س ت ه ت ن ی ھ ایای ن در ھ س ت ن د. ب رخ وردار وی ژه ای اھ م ی ت از ھ م وردا ت واب ج ا ای ن در م ش ون د. م ع رف م ق دار م ات ری س ھ س ت ه ھ ای از ج دی د رده ی م ق ال ه ای ن درپ ردازش ن ظ ری ه در ت واب ای ن م ش ود. ت ب دی ل خ روج ف ض ای روی م ت ن اظ ر گ روه ع م ل ھ ای ب ه ت اب ورودی ف ض ای روی گ روه ع م ل ھ ای ت واب پ االی ش ک ه ھ س ت م ع ن ای ن “ب ه ن اوردا زم ان‐ ” واژه ھ س ت ن د. م ش ه ور “ زم ان‐ن اوردا خ ط پ االی ش ” ع ن وان ب ه ت ص وی ر پ ردازش و س ی ن الپ االی ش ھ م و زم ان ان ت ق ال گ روه ن م ای ش ھ ای ک ه م ک ن د ب ی ان زم ان ان ت ق ال ھ م وردای دی ر ع ب ارت ب ه ھ س ت. ھ م وردا زم ان، ان ت ق ال گ روه ب ه ن س ب ت

م ش ود. ت وص ی ه [٢ ،١] م راج م ط ال ع ه راس ت ا ای ن در ک ه ھ س ت ن د گ روه روی ان ت رال گ ی ری اس اس ب ر ھ س ت ه ھ ا م ش ون د. ج ا ب ه ج ا

گ روه ھ ا روی ان ت رال گ ی ری ٢

ب ع د ب ا ھ ی ل ب رت ف ض ای ی V ک ه اس ت G 7→ L(V) ھ م ری خ ت ی ρg گ روه ن م ای ش ب ی ری د. ن ظ ر در را G ت پ ی م ان ه ای و خ ط ف ش رده، گ روهع م ل ھ ا ( اس ت ل گ روه ی G ) پ ی وس ت ه ح ال ت در اس ت. x ∈ V ک ه م ش ود داده ن م ای ش gx ب ا را ρgx راح ت ب رای اس ت. م ت ن اھ ای ن در ک ه ک رد م ع ط وف ی ان گ روه ع م ل ھ ای روی را خ ود ت وج ه ک ل ی ت دادن دس ت از ب دون م ت وان ح ال ت ای ن در ب اش ن د. پ ی وس ت ه ب ای دf : X → Y ت واب .[٣] م ش ود داده ن م ای ش

∫G f(g)d(g) ص ورت ب ه ھ ار ان ت رال اس ت. ب رق رار ρg−١ = ρ†

g راب ط ه ھ م واره گ روه ھ اداده ن م ای ش ⊗ ع الم ت ت وس ط ک رون ه ک ر ض رب ھ ای ی ا ت ان س ور ھ س ت ن د. م ت ن اھ ب ع د ب ا ھ ی ل ب رت ف ض اھ ای Y و X ک ه ھ س ت ن د ن ظ ر م وردY و X در گ روه ع م ل ھ ای ک ن ی د ف رض م ک ن د. ص دق f(gx) = gf(x) راب ط ه در g ∈ G و x ∈ X ھ ر ب رای ھ م وردا ت اب ی م ش ون د.ی ع ن ن اورداس ت داخ ل ض رب ی ان گ روه ن م ای ش ب ه ت وج ه ب ا م ش ون د. داده ن م ای ش ⟨. | .⟩X ص ورت ب ه داخ ل ض رب ھ ای ب اش ن د. م ن اس ب

ھ م چ ن ی ن اس ت. k : X × X → R م ان ن د م ت ق ارن م ث ب ت، م ع ی ن ت اب ی اس ال ر م ق ادی ر ب ا ھ س ت ه ی .⟨x١ | gx٢⟩X = ⟨x١ | x٢⟩Xت وس ط ش ده ت ول ی د ت اب ع ف ض ای .k(gx١, gx٢) = k(x١,x٢) ،g ∈ G ھ ر ب رای ی ع ن ھ س ت ن د، ن اوردا G گ روه ھ ای ع م ل م ش ود ف رضی ع ن ھ س ت ن د، Y در و م ق دار ب ردار ض ری ب ھ ا ک ه م دھ ن د ن م ای ش Zk ب ا را k ھ س ت ه ھ ای از ب رداری م ق ادی ر ب ا خ ط ت رک ی ب ات ت م ام

ب ا ،f(xi) = yi ک ه (xi,yi) : i = ١, ..., n ن م ون ه ھ ای از f : X → Y ت اب .Zk = f(x) =∑

i k(x,xi)ai|xi ∈ X ,ai ∈ Yھ س ت زی ر ص ورت ب ه Zk ک م ی ن ه م ق دار ب ی ری د. ن ظ ر در را ھ م وردای خ اص ی ت

f(x) =

n∑

i=١k(x,xi)ai (١۵)

١٨٩

م ه دی ش م س،

م ش ون د. اخ ت ی ار (gxi, gyi)|i = ١, ..., n, g ∈ G ص ورت ب ه ن م ون ه ھ ای ھ م وردا رف ت ار ی آوردن دس ت ب ه ب رای .ai ∈ Y آن در ک هص ورت ب ه ج واب ھ ا ب ن اب رای ن

f(x) =

∫

G

n∑

i=١k(x, gxi)gai dg, (١۶)

م ش ون د. ن وش ت ه

اس ت. G ب ه ن س ب ت ھ م وردا ت اب ی (٢) ص ورت ب ه ت اب ھ ر .٢ . ١ ل م

گ ف ت م ت وان ھ س ت ه ن اوردای ب ه ت وج ه ب ا اث ب ات.

f(gx) =

∫

G

n∑

i=١k(x, g−١g′xi)g

′ai dg′,

:h = g−١g′ ص ورت ب ه ان ت رال ک ردن پ ارام ت ری دوب اره ب ا G ب ودن ت پ ی م ان ه ای ب ه ت وج ه ب ا و

f(gx) =

∫

G

n∑

i=١k(x, hxi)ghai dh = gf(x).

ک رد: ب ازن وی س زی ر ص ورت ب ه را f م ت وان (٢) در ج م و ان ت رال ج ا ب ه ج ای ب ا

f(x) =n∑

i=١

(∫

Gk(x, gxi)ρgdg

)ai (١٧)

ش ود. گ رف ت ه ن ظ ر در م ق دار م ات ری س ھ س ت ه ی ع ن وان ب ه م ت وان د پ اران ت ز داخ ل م ق دار ک ه

م ش ود ت ع ری ف K(x١,x٢) =∫G k(x١, gx٢)ρgdg ص ورت ب ه x١,x٢ ∈ X ھ ر ب رای K : X × X 7→ L(Y) ک ن ی د ف رض .٢ . ٢ گ زاره

ب ن اب رای ن اس ت. م ت ق ارن G‐ن اوردای ت اب ی k و گ روه ن م ای ش ی ρg ∈ L(Y) آن در ک ه

،g, h ∈ G و x١,x٢ ∈ X ھ ر ب رای ال ف)K(gx١, hx٢) = ρgK(x١,x٢)ρ†

h

اس ت. ھ م وردا K م ش ود گ ف ت ه ح ال ت ای ن در ک ه اس ت ن اھ م وردا م ول ف ه دوم ی ن در و ھ م وردا م ول ف ه اول ی ن در K ی ع ن

.K(x١,x٢) = K(x٢,x١)† ب)

G ب ودن ت پ ی م ان ه ای و k ب ودن م ت ق ارن و ن اوردای خ اص ی ت ب ه ت وج ه ب ا (ب) ب رای و م ش ود اس ت ف اده ٢ . ١ ل م از (ال ف) اث ب ات ب رای اث ب ات.ن وش ت: م ت وان

K(x١,x٢) =

∫

Gk(g−١x١,x٢)ρgdg =

∫

Gk(x٢, gx١)ρg−١dg = K(x٢,x١)†.

در داخ ل ض رب ی ھ س ت ه ک ه دارد وج ود Φ : X → F م ش خ ص ه ت اب ی F م ش خ ص ه ف ض ای ی ب اش د، اس ال ر ھ س ت ه ی K اگ رش ود. م ع رف زی ر ص ورت ب ه م ت وان د ای ده ای ن م ق دار م ات ری س ھ س ت ه ھ ای ب رای .K(x١,x٢) = ⟨Φ(x١) | Φ(x٢)⟩F ی ع ن اس ت، ف ض ا ای ن

،x١,x٢ ∈ X ھ ر ب رای ھ رگ اه اس ت م ق دار م ات ری س ھ س ت ه ی K : X × X 7→ L(Y) ت اب م ق دار) م ات ری س (ھ س ت ه .٢ . ٣ ت ع ری ف

K(x١,x٢) = ⟨Ψ(x١) | Ψ(x٢)⟩H

اس ت. م ث ب ت م ع ی ن ⟨Ψ | Ψ⟩H ∈ L(Y) م ات ری س ،٠ = Ψ ∈ H ھ ر ب رای ک ه ھ س ت خ اص ی ت ای ن ب ا Ψ : X 7→ H = F ⊗ L(Y) ک ه

190

م ق ادی ر م ات ری س ھ س ت ه ھ ای ب ا ھ م وردا ت واب

ھ س ت ه را آن ک ه اس ت م ق دار م ات ری س ھ س ت ه ی ش ده م ع رف ٢ . ٢ گ زاره در ک ه K ت اب ب اش د اس ال ر ھ س ت ه k اگ ر م ش ود، م ش اھ ده راح ت ب هگ وی ن د. گ روه ان ت رال گ ی ری م ات ری س

ص ورت ب ه ΠE ی ان ت ص وی ر اس ت. خ ط ف ض ای زی ر ی ھ م وردا، ت واب از E ⊂ Zk ف ض ای زی ر ک رد ث اب ت م ت وان راح ت ب ه

(ΠEf)(x) =١

µ(G)

∫

Gg−١f(gx) dg

زی را: دارد (٢) ف رم ب ه ن م ای ش (١٨) از ΠE ت ص وی ر م ش ود. ت ع ری ف

(ΠEf)(x) =١

µ(G)

∫

G

n∑

i=١k(gx,xi)g

−١aidg

=١

µ(G)

∫

G

n∑

i=١k(x, g−١xi)g

−١ai dg

=١

µ(G)

∫

G

n∑

i=١k(x, hxi)hai dh

اس ت. ش ده گ رف ت ه ن ظ ر در h = g−١ آن در ک ه

ص ورت ای ن در ب اش د، زی ان ت اب c : (X × Y × Y)n 7→ R و ص ع ودی اک ی دا ت اب ی Ω : R+ 7→ R اگ ر ھ م وردا) (ن م ای ش ر .۴ . ٢ ق ض ی هم ق دار ک ه f ∈ Zk ھ م وردای ت اب ھ ر

R(f) = c(x١,y١, f(x١), ...,xn,yn, f(xn)) + Ω(∥ f ∥٢)

ص ورت ب ه م ک ن د ک م ی ن ه را

f(x) =

∫

G

n∑

i=١k(x, gxi)gai dg

اس ت.

ھ م وردا ت اب ھ ر ،i = ١, ..., n ،f⊥(xi) = ٠ س اخ ت ن ت وس ط ش ود، ت ج زی ه f(x) = f∥(x) + f⊥(x) ص ورت ب ه f ∈ Zk ت اب اگ ر اث ب ات.ب ن اب رای ن اس ت، ΠEf از ت ص وی ر ی Zk در

(ΠEf)(x) = (ΠEf∥)(x) + (ΠEf⊥)(x).

ک ه ای ن از و م م ان د ت غ ی ی ر ب ،ΠEf⊥ از ک ردن ن ظ ر ص رف ش رط ب ه R(ΠEf) در زی ان ت اب ب ن اب رای ن .(ΠEf⊥)(xi) = ٠ ھ م چ ن ی ن

ΠEf∥ =

∫

G

n∑

i=١k(x, gxi)gai dg

ب رای رو ای ن از .Ω(∥ ΠEf ∥٢) ≥ Ω(∥ ΠEf∥ ∥٢) و ∥ ΠEf ∥٢=∥ ΠEf∥ ∥٢ + ∥ ΠEf⊥ ∥٢ ب ودن، م ت ع ام د خ اص ی ت ب ه ت وج ه ب ا وم ش ود. ک م ی ن ه ن ظ ر م ورد م ق دار ΠEf⊥ = ٠ ان ت خ اب ب ا و R(ΠEf) ≥ R(ΠEf∥) ن وش ت م ت وان ai ث اب ت م ق ادی ر

م راج

[1] H. Burkhardt and S. Siggelkow. Invariant features in pattern recognition - fundamentals and applications. InNonlinear Model-Based Image/Video Processing and Analysis, pages 269–307. John Wiley and Sons, 2001.

[2] B. Haasdonk, A. Vossen, and H. Burkhardt. Invariance in kernel methods by haar-integration kernels. In Pro-ceedings of the 14th Scandinavian Conference on Image Analysis, pages 841–851, 2005.

[3] L. Nachbin. The Haar Integral. D. van Nostrand Company, Inc., Princenton, New Jersey, Toronto, New York,London, 1965.

[4] B. Schoelkopf and A. J. Smola. Learning with Kernels. The MIT Press, 2002.

[5] J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, 2004.

191

م زاح م پ ارام ت ر ب ا م دل ی در ب راورد م ع ادل ه ی اف ت ن ب رای گ روه ع م ل ک ارب ردش م س* م ه دی

ای ران) ک اش ان، ک اش ان، (دان ش اه

ت ب دی ل ن م ون ه ف ض ای روی ت ب دی الت از G ت وپ ول وژی گ روه ی ت وس ط P٠ ث اب ت اح ت م ال ان دازه ی س اده، ت ب دی ل م دل ی در چ ی ده:ش ود، آغ از Pψ م ان ن د دی ری م دل از P٠ ج ای ب ه ت ب دی ل اگ ر ش ود. م گ ذاری ان دی س g ∈ G پ ارام ت ر ت وس ط م دل ی رو ای ن از و م ش وددارد ب س ت ψ ب ه ت ن ه ا ک ه اس ت ت وزی ع دارای آم اره ای ن ش ود. م ح ذف گ روه ت ح ت پ ی ش ی ن ن اوردای آم اره ت وس ط س ازی ک ن اری ب ا گ روه پ ارام ت رψ ب ه ن ی ز ب ی ش ی ن ن اوردای آم اره آن درپ و گ روه ک ه ش ود م م رت ف زم ان م ش ل ش ود. م ح اص ل ψ روی اس ت ن ب اط ب رای پ ای ه خ وب رو ای ن از وم ش اھ دات م ه ح ال ت در ψ ب رای ب رآورد م ع ادل ه ی س اخ ت ن ب رای گ روه س اخ ت ارھ ای اس ت ف ادھ از چ ون م ق ال ه ای ن در ب اش د. داش ت ه ب س ت

ش ود. م داده ن ش ان ان د ش ده اس ت خ راج ث اب ت ب ا ھ ای ت وزی از م س ت ق لق س م ت خ ارج ان دازه ن اوردا، ن س ب ت ا ان دازه چ پ، ن اوردای ان دازه ، ن اوردای گ روه، ع م ل ب راورد، م ع ادل ه ک ل ی دی: ک ل م ات

١٠F۶٢ ،١١H۵۴ م وض وع:

م ق دم ه ١

م دل ی ای ن ک ن ی د ف رض ھ س ت ن د. م ج ه ول پ ارام ت رھ ای g و ψ آن در ک ه ب اش د Pψ,g ت وزی ب ا E ⊆ Rn روی ت ص ادف م ت غ ی ر ی Y ک ن ی د ف رضھ ر ب رای ک ه γg : X → X ت ب دی الت وس ی ل ه ب ه ن م ون ه ف ض ای روی ک ه ھ س ت ت وپ ول وژی گ روه ی G آن در ک ه ب اش د g ∈ G پ ارام ت ر ب ا ت ب دی لآغ از Pψ = Pψ,e ت وزی از ک ه ت ب دی الت ت وس ط م دل ،Pψ,g = γg(Pψ) گ رف ت ن ن ظ ر در ب ا م ک ن د. ع م ل ،γgg′ = γg γ′

g ،g, g′ ∈ G

اس ت. م زاح م پ ارام ت ر g ھ م چ ن ی ن و ψ ع الق ه م ورد پ ارام ت ر اس ت. G گ روه ھ م ان ع ض و e آن در ک ه م ش ود ال ق ا م ش ون د،ψ ب ه راج اس ت ن ب اط ب رای آم اره ای ن و دارد ب س ت ψ ب ه ت ن ه ا ک ه اس ت ت وزی ع دارای ت ب دی الت گ روه ت ح ت T (Y ) ب ی ش ی ن ن اوردای آم ارهب ه م ورد ای ن در م ط ال ب م ش اھ ده ب رای م ش ود. ان ج ام Y |T (Y ) ش رط ت وزی روی g ب ه راج اس ت ن ب اط ع م وم ا ک ه ص ورت در اس ت، م ن اس بم ق ی اس پ ارام ت ر و ψ ش ل پ ارام ت ر ب ا گ ام ا ت وزی ی از م س ت ق ل م ول ف ه ھ ای Y = (Y١, . . . , Yn) اگ ر م ث ال ع ن وان ب ه ش ود. م راج ع ه [٧ ،۶]ل ذا و ن دارد ب س ت ψ ب ه گ روه ع م ل م ث ال ای ن در اس ت. ب ی ش ی ن ن اوردای T = (Y١/Y , . . . , Yn/Y ) آم اره ،y 7→ gy ت ب دی ل ب ا ب اش د، g > ٠ک ام ال g م زاح م پ ارام ت ر ت اث ی ر آن پ در و م ش ود اس ت ف اده T ک ن اری ت وزی از ψ ب ه راج اس ت ن ب اط ب رای اس ت. واب س ت ه ψ ب ه ت ن ه ا ن ی ز T آم اره ت وزی پ رداخ ت ه ψ ب راورد ب رای گ روه س اخ ت ارھ ای از اس ت ف اده چ ون ب ررس ب ه [٨] و [۵ ،۴ ،٣ ،٢] ت ح ق ی ق ات پ ی رو م ق ال ه ای ن در م رود. ب ی ن از

م ش ود.

ب ی ش ی ن ن اوردای ت وزی ٢

و ھ س ت ن د X روی ھ م ارزی رده ھ ای م دارھ ا، اس ت. γg(y) : g ∈ G ص ورت ب ه م ج م وع ه ی G ع م ل ت ح ت ن م ون ه ف ض ای در م دار ی م دار ب ه را y ،π : X → X/G م داری ت ص وی ر م دھ ن د. ن م ای ش X/G ن م اد ب ا را آن ک ه اس ت ق س م ت خ ارج ف ض ای ی م دارھ ا م ج م وع ه

اس ت. π(y) از ت اب ع ن اوردا آم اره ھ ر و اس ت G ع م ل ت ح ت ب ی ش ی ن ن اوردای آم اره ی ت اب ای ن و م ک ن د ت ص وی ر آن ب ا م ت ن اظ رخ ارج ان دازه ،X/G روی ان دازه اس ت. X/G روی ان دازه ی ب ه ن س ب ت چ ال ی ع ن وان ب ه ب ی ش ی ن ن اوردای ت وزی ت وص ی ف ھ دفف ش رده، م ج م وع ه ھ ر وارون ت ص وی ر ی ع ن ب اش د، م ن اس ب (g, y) 7→ (γg(y), y) ت اب ی ع ن گ روه ع م ل ک ن ی د ف رض .( [١] (ر.ک. اس ت. ق س م ت

١٩٢

م زاح م پ ارام ت ر ب ا م دل ی در ب راورد م ع ادل ه ی اف ت ن ب رای گ روه ع م ل ک ارب رد

ان دازه ی λ و دارد) وج ود ف ش رده م وض ع ا گ روه ی روی ان دازه ای چ ن ی ن (اغ ل ب G روی راس ت ن اوردای ان دازه β ک ن ی د ف رض ب اش د. ف ش ردهی ع ن ب اش د، ∆−١

G ض رب ک ن ن ده ب ا X روی ن اوردا ن س ب ت ا

γg(λ) = ∆(g)λ (١٨)

ح ال ت در ،∆(g١.g٢) = ∆(g١)∆(g٢) ،g١, g٢ ∈ G ھ ر ب رای ای ن ه ب ه ت وج ه ب ا اس ت. گ روه پ ی م ان ه ای ت اب ∆ : G → R+ آن در ک هsupp(βw) = π−١(w) ک ه X روی (βw)w∈X/G ان دازه ھ ای و X/G روی λ/β ق س م ت خ ارج ان دازه ص ورت ای ن در .∆(e) = ١ خ اص

،X روی h ان ت رال پ ذی ر ت اب ھ ر ب رای ی ع ن اس ت، π ب ه ن س ب ت λ از ت ج زی ه ی (βw)w∈X/G, λ/β ک ه گ ون ه ای ب ه دارن د وج ود∫

Xh(y)dλ(y) =

∫

X/G

(∫

π−١(w)h(y)dβw(y)

)d(λ/β)(w).

ص ورت ب ه (βw)w∈X/G ∫ان دازه ھ ای

π−١(π(z))h(y)dβπ(z)(y) =

∫

Gh(γg(z))dβ(g)

م ش ون د. داده z ∈ X ک هگ ون ه ای ب ه X روی η م ث ب ت ت اب ی اس ت. χ٠ ض رب ک ن ن ده ب ا ن اوردا ن س ب ت ا ک ه ب اش د µ ان دازه ب ه ن س ب ت f چ ال دارای Y ک ن ی د ف رضھ ر ب رای π(Y ) ب ن اب رای ن م ک ن د. ص دق (١٨) راب ط ه در λ = (١/η)µ ان دازه ، ت اب ای ن ب ا و η(γg(y)) = χ٠(g)∆(g)η(y) ک ه دارد وج ود

چ ال ت اب دارای y ∈ π−١(w)

k(w) = η(y)

∫

Gf(γg(y))χ٠(g)∆(g)dβ(g)

اس ت. G روی چ پ ن اوردای ان دازه ی ∆(g)dβ(g) ان دازه اس ت. λ/β ق س م ت خ ارج ان دازه ب ه ن س ب ت w ∈ X/G در

ع الق ه م ورد پ ارام ت ر ب رای ب راورد م ع ادل ه ٣

ھ ر ب رای ی ع ن ب اش د ن ااری ب م ع ادل ه اگ ر ن ظ م، ش رای ط ت ح ت ب ی ری د. ن ظ ر در را اس ت∑n

i=١ h(ψ, yi) = ٠ ص ورت ب ه ک ه ψ ب رای ب راورد م ع ادل هاس ت. س ازگ ار ψ ب راورد ،Eψ,gi

h(ψ, Yi) = ٠ ،ψ, giی ع ن ام ت ی از ت اب آن پ در و l(ψ, g; y) = log f(y;ψ, g) ی ع ن ب اش د، واح د م ول ف ه ی از درس ت ن م ای ت اب ل اری ت م ،l ک ن ی د ف رض

م ش ود: ت ع ری ف زی ر ت اب اک ن ون م ش ون د. م ح اس ب ه lg(ψ, g; y) = ∂∂g l(ψ, g; y) و lψ(ψ, g; y) = ∂

∂ψ l(ψ, g; y)

m(ψ, g;π٠) = Eψ,glψ(ψ, g;Y )|π(ψ;Y ) = π٠.

زی را اس ت ص ف ر ت اب ای ن م ی ان ی ن

Eψ,gm(ψ, g;π(ψ;Y )) = Eψ,gEψ,glψ(ψ, g;Y )|π(ψ;Y )= Eψ,g lψ(ψ, g;Y )

= ٠.

ب ا ξ(πi) م ث ل πi ت اب ھ ر ب رای ح ق ی ق ت ای ن اس ت. ع م ود gi م ق دار ھ ر در ly ل ذا دارد، ب س ت πi ط ری ق از yi ب ه ت ن ه ا m(ψ, g;πi) ھ م چ ن ی نزی را اس ت، ص ح ی ص ف ر م ی ان ی ن

Eψ,giξ(πi)lg(ψ, gi;Yi) =

∂

∂giEψ,gi

ξ(πi) = ٠

از ن ااری ب ب راورد ی ب ت وان ث اب ت ψ ھ ر ب رای ک ن ی د ف رض ن دارد. ب س ت gi ب ه م ی ان ی ن ک ه اس ت ش ده ص ف ر ب راب ر دل ی ل ای ن ب ه آخ ر ت س اوی ک ه:π٠ و g ،ψ ھ ر ب رای ک ه ب اش د داش ت ه وج ود گ ون ه ای ب ه h(ψ; yi) ت اب ی ع ن ی اف ت، πi ش رط ب ه ش رط ت وزی از m(ψ, g;πi)

Eψ,gh(ψ;Yi)|π(ψ;Yi) = π٠ = m(ψ, g;π٠).

ک ه ای ن ب ه ت وج ه ب ا اس ت. ب راورد م ع ادل ه ب رای پ ی ش ن ه ادی∑n

i=١ h(ψ; yi) ب ن اب رای ن

Eψ,gih(ψ;Yi)) = Eψ,gi

Eψ,gih(ψ;Yi)|π(ψ;Yi)

= Eψ,gim(ψ, gi;π(ψ;Yi))

= ٠,

193

م ه دی ش م س،

ھ م چ ن ی ن: اس ت، ن ااری ب م ع ادل ه

V arψ,gih(ψ;Yi) = V arψ,gi

m(ψ, gi;π(ψ;Yi)) + Eψ,giV arψ,gi

h(ψ;Yi)|π(ψ;Yi).

ع الق ه م ورد پ ارام ت ر σ ک ه ب اش ن د σ اس ت ان دارد ان ح راف و µi م ی ان ی ن ب ا م س ت ق ل ن رم ال ت ص ادف م ت غ ی رھ ای Yi٢ و Yi١ ک ن ی د ف رض .٣ . ١ م ث ال

σ٢ ب رای ام ت ی از ت اب اس ت. ب ی ش ی ن ن اوردای آم اره Di ک ه م ش ود م ش اھ ده راح ت ب ه .µi =Yi١ + Yi٢

٢و Di = Yi٢ − Yi١ دھ ی د ق رار اس ت.

ب ا اس ت ب راب ر م ش اھ ده ج ف ت n از ح اص ل

∑

i

lσ٢(σ٢, µi;Yi) = − n

σ٢ +١σ۴

(∑iD

٢i

۴+∑

i

(µi − µi)٢)

σ٢

٢ ب ه اح ت م ال در ب راورد ای ن و اس ت − n٢σ٢ م ی ان ی ن دارای ک ه م آی د دس ت ب ه − n

σ٢ + ١۴σ۴∑D٢i م ق دار µi ج ای ب ه µi دادن ق رار ب ا ک ه

اس ت. ھ م را

از اس ت ع ب ارت ب ی ش ی ن ن اوردای آم اره ش رط ب ه lσ٢ ش رط م ی ان ی ن

m(σ٢, µi;Di) = − ١٢σ٢ +

١۴σ۴D

٢i

اس ت. µ١, ..., µn از م س ت ق ل ک ه م ش ود م ن ج ر − n٢σ٢ + ١

۴σ۴∑

iD٢i = ٠ ب راورد م ع ادل ه ب ه ک ه

م راج

[1] S. Andersson, (1982), Distributions of Maximal Invariants using Quotient Measures, Ann. Statist., 10(3), 955-961.

[2] D. A. S. Fraser, (1967), Data transformations and the linear model, Annals of Mathematical Statistics, 38, 1456-1465.

[3] D. A. S. Fraser, (1968), The Structure of Inference, John Wiley Sons, Inc., New, York-London-Sydney.

[4] D. A. S. Fraser, (1972), The determination of likelihood and the transformed regression model, Annals of Math-ematical Statistics, 43, 896-916.

[5] D. A. S. Fraser, (1979), Inference and linear models,McGraw-Hill, New York.

[6] J. R. Gabriel and S. M. Kay, (2002), Use of Wijsman’s theorem for the ratio of maximal invariant densities insignal detection applications, Conference Record of the Thirty-Sixth Asilomar Conference on Signals, Systemsand Computers.

[7] P. Majerski, and Z. Szkutnik, (2010), Approximations to most powerful invariant tests for multinormality againstsome irregular alternatives, Test, 19(1), 113–130.

[8] J. Neyman, and E. L. Scott, (1948), Consistent estimates based on partially consistent observations, Economet-rica, 16, 1-32.

194

Abstracts of the Talks in Workshop(In Alphabetical Order)

Abstracts of the Talks in Workshop

Statistical Instability for Rovella Maps

Mohammad Alikhan (IPM)

Abstract: We consider a one parameter family of one-dimensional maps introduced by Rovella, ob-tained through modifying the eigenvalues of the geometric Lorenz attractor, replacing the expanding con-dition on the eigenvalues of the singularity by a contracting one. We show that the Rovella maps are notstatistically stable on a set of parameters having physical measure.

Entropy formula and continuity of entropy forpiecewise expanding maps

Jose Alves (Porto University, Portugal)

Abstract:Abstract: We consider some classes of piecewise expanding maps in finite dimensional spaceswith invariant probability measures, which are absolutely continuous with respect to Lebesgue measure. Wederive an entropy formula for such measures. Using this entropy formula, we present sufficient conditionsfor the continuity of that entropy with respect to the parameter in some parametrized families of maps. Weapply our results to some families of piecewise expanding maps. Joint work with Antonio Pumario.

This is the one with two lectures? If you want two different titles, consider one with the entropy formulaand the other one with continuity of entropy.

196

Abstracts of the Talks in Workshop

Rates of mixing for discrete dynamical systems

Peyman Eslami (Warwick University, UK)

Abstract: After a general overview of dynamical systems, I will focus on the question of mixing ratesin chaotic systems. Using some the most modern and flexible techniques, I will show how to answer thisquestion for various dynamical systems starting with uniformly expanding maps.

Statistical stability for contracting Lorenz flows

Mohammad Soufi (Universidade Federal do Rio de Janeiro, Brazil)

Abstract: The contracting Lorenz flow is a geometric Lorenz flow whose orbits spend much more longertime than normal Lorenz flow near the stable manifold of its equilibrium point. Reducing the contractingLorenz flow to a one-dimensional map yields a Lorenz-like map whose discontinuity point coincides with itscritical point. Using the M. Benedicks and L. Carleson’s result on quadratic map, A. Rovella proved that thecontacting Lorenz flow supports a strange attractor which it is not persistent under C3 perturbations of theflow. But there is a codimensional submanifold in the space of all vector fields such that its intersection pointwith a two-dimensional transversal submanifold is a density point of vector fields with strange attractors.It was proved that each strange attractor of contracting Lorenz flow supports a unique SRB measure. Inthis talk we are going to show that these SRB measures vary continuously in weak-star topology. This is akind of stability called statistical stability.

197

department of mathematics, faculty of mathematical...

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