FUNCTION ALGEBRAS OVER GROUPS

V. MURUGANANDAM

Abstract. One of the primary aspects of harmonic analysis isthe study of functions on a homogenous space by means of suitable

group actions. The other end of this thread is to introduce variousclasses of functions on groups and study their properties vis-a-visthe underlying groups. In this compact course, we focus on thisconcept with particular reference to the most useful and widelystudied notion in harmonic analysis namely �Amenability�. Weaim to � look beyond the amenability� at the end of the course.

1. Preliminaries

De�nition 1.1. A topological space G is said to be a topological groupif it is a group and the map (x, y) → xy−1 is continuous from G × Ginto G.

Let us give some important classes of locally compact Hausdor�groups.

Examples 1.2. (i) Rn is a locally compact group.(ii) For every n ≥ 1, Tn is a compact group.(iii) For every �xed n, let GL(n,R) be the group consisting of all

n× n invertible matrices with real entries. Then GL(n,R) is a locallycompact group as it is an open subset of Rn2

.(iv) SL(n,R) = {x ∈ GL(n,R) : det(x) = 1} and O(n) consisting of

orthogonal matrices are also locally compact groups as they form closedsubsets of GL(n,R).

(v) S =

{x =

(a b0 1

): a > 0, b ∈ R

}.

(vi) If G is the Heisenberg group given byg =

1 x z0 1 y0 0 1

: x, y, z ∈ R

0This notes is based on a series of lectures given in the Workshop and a centenary

conference on Analysis and Applications, IISc Mathematics Initiative (IMI), I.I.Sc,Bangalore, from May, 14-23, 2009. The author thanks R. Lakshmi Lavanya forwriting down the notes.

1

2 V. MURUGANANDAM

then G is noncompact and nonabelian.(vii) Let F2 denote the free group generated by a, b, with ab 6= ba. It

is a nonabelian discrete group. Arbitrary element of this group is of theform ambnak · · · where m,n, k belong to Z.

Let Cc(G) and Cb(G) denote the space of all continuous functionswith compact support and continuous bounded functions on G, respec-tively.Let Clu(G), Cru(G) and Cu(G) denote the space of all left uniformly

continuous functions, right uniformly continuous functions and uni-formly continuous functions on G.

1.1. Measure algebra.

De�nition 1.3. Let X be a locally compact space. A Borel measure µis said to be regular if

(1) µ(K) <∞, for every compact set K.(2) For every Borel set E, µ(E) = inf{µ(U) : U is open and E ⊆

U }.(3) If E is any Borel set and µ(E) <∞, then µ(E) = sup

{µ(K) :

K is compact, K ⊆ E}.

De�nition 1.4. A Banach space A over C is called a Banach algebraif A is an algebra satisfying

‖xy‖ ≤ ‖x‖‖y‖, ∀ x, y ∈ A

and is called Banach?-algebra if it admits an involution x → x? on Asuch that ‖x?‖ = ‖x‖ for all x in A.

Let G be a locally compact Hausdor� group. If M(G) denotes thespace of complex Borel measures on G, then it forms a Banach Space.In what follows, we brie�y show that the underlying group structuregives rise to two additional structures on M(G), with respect to whichM(G) forms an Banach ? algebra. If µ, ν are in M(G), then de�ne∫

φ(x) d(µ ∗ ν)(x) =

∫ ∫φ(xy) dµ(x) dν(y).

µ∗ν is called the convolution of the measures. For every µ and ν, µ∗νbelongs to M(G) and ‖µ ∗ ν‖ ≤ ‖µ‖ ‖ν‖ .Recall that if x belongs to G, then δx denotes the Dirac measure at

x. If x, y are in G, then∫φ d(δx ∗ δy) =

∫ ∫φ(uv) dδx(u) dδy(v) = φ(xy) =

∫φ dδxy.

3

Therefore, δx ∗ δy = δxy. See that δe is the multiplicative identity ofM(G). The involution µ→ µ? on M(G) is de�ned by∫

φ dµ? =

∫φ(x−1) dµ(x).

It is easy to see that δe is the identity for M(G).

1.2. Group algebra. One of the milestones in the history of abstractHarmonic analysis was the existence of left invariant measure on ageneral locally compact Hausdor� group that reads as follows. A Borelmeasure m on G is said to be left invariant if m(xE) = m(E) for everyBorel set E, and for all x in G.

Theorem 1.5. (Haar, Von Neumann) Let G be a locally compactHausdor� group. There exists a non-zero, positive, left invariant, regu-lar Borel measure on G.Moreover, it is unique up to a positive constant.

Such a measure is called Haar measure and the corresponding in-tegral is called Haar integral. We refer to the books by A. Weil[18],Loomis[11] for a proof. Let dx always denote the Haar integral.Let us see some examples of Haar measures.

(1) For any subset E of a discrete group, if m(E) is de�ned to bethe number of elements in E, then it is a Haar measure calledcounting measure.

(2) The Lebesgue measure is the Haar measure for the group Rn,as it is translation invaraint.

(3) For the Torus T, 12π

∫ 2π

0f(eıx)dx is the Haar integral.

(4) If G = GL(n, R) then verify that 1(det g)n

dg11dg12 · · · dgnn de�nesa Haar measure.

(5) If G denotes the group given in 1.2(v) then 1a2dadb is the Haar

measure.(6) The Lebesgue measure

dg = dxdydz

is the Haar measure for the Heisenberg group given in 1.2(v).

Let Lp(G), 1 ≤ p <∞ denote as usual, the Banach space consistingof all measurable functions f such that

∫G|f(x)|p dx is �nite. Similarly

one de�nes L∞(G).

Theorem 1.6. (Lusin). For every p, 1 ≤ p < ∞, Cc(G) is dense inLp(G).

Proposition 1.7. If f in Lp(G), 1 ≤ p < ∞ is �xed then the mapx→x f from G into Lp(G) is continuous.

4 V. MURUGANANDAM

Proof. Let us assume that f ∈ Cc(G). Then, f is uniformly continuous.Let K be the support of f. Let ε > 0 be given. Since G is locallycompact, there exists a neighborhood W of identity e such that W iscompact. Choose a neighborhood U of e such that U ⊆ W and if x, y ∈G are such that x ∈ Uy then

|f(x)− f(y)| <√

ε

µ(KW ),

µ being the �xed Haar measure.Then for every x, y ∈ G such that x ∈ Uy , we have ||xf −y f ||p < ε.Use Lusin's theorem to show that the result holds for arbitrary ele-

ment of Lp(G). �

A Haar measure need not be right invariant. In fact, for a �xed xif we de�ne dmx by 〈 f, dmx 〉 =

∫Gf(yx)dy then one can see that

it is left integral. Therefore, there exists a function x → ∆(x) calledmodular function satisfying the following:∫

G

f(yx)dy = ∆(x−1)

∫G

f(y)dy

The modular function of G can easily be seen to be a homomorphismfrom G into R+. By Proposition 1.7, we see that ∆ is continuous.A group G is called Unimodular if

∆(x) = 1 ∀x ∈ G.

It is trivial that any abelian group or discrete group is unimodular. As∆ is a continuous homomorphism, any compact group is unimodular.In fact all groups enumerated in 1.2 except the group S in example (v)

are unimodular. ∆(x) = a−1, if x = x =

(a b0 1

).

Recall that by Radon-Nikodym theorem we can view L1(G) as aclosed subspace ofM(G) consisting of all measures which are absolutelycontinuous with respect to Haar measure. We show that L1(G) is aBanach ? subalgebra of M(G).For every f, g ∈ L1(G), since∫

G

φ(x)d(f ∗ g)(x) dx =

∫G

∫G

φ(xy)f(x)g(y) dx dy

=

∫G

∫G

φ(x)f(y)g(y−1x) dy dx,

5

we have

f ∗ g(x) =

∫G

f(y)g(y−1x) dy. (1.1)

Similarly, the involution of M(G), restricted to L1(G), is given by

f ?(x) dx = f(x−1) d(x−1).

That is,

f ?(x) = ∆(x−1)f(x−1). (1.2)

Summarizing we observe that with the convolution product (1.1) andthe involution (1.2), L1(G) forms an Banach ?-algebra called groupalgebra. Moreover it is a two-sided ideal in M(G) and closed underinvolution. In fact

µ ∗ g(x) =

∫G

g(y−1x) dµ(y).

f ∗ µ(x) =

∫G

∆(y−1)f(xy−1) dµ(y).

One can easily see that if G is commutative then L1(G) is a com-mutative Banach?-algebra. With a little more e�ort, one can prove theconverse also. That is, if L1(G) is commutative with the convolutionproduct then the underlying group is commutative.Let us also remark that if G is discrete then δe is the identity for

L1(G). We recall that the converse also holds. That is L1(G) has iden-tity if and only if G is discrete.But nevertheless, the group algebra of a general locally compact

group, has bounded approximate identity. We brie�y construct onesuch as follows.Let {Vα}α∈ I be a neighbourhood system consisting of compact neigh-

bourhoods at e. Set

fα =gα

µ(Vα), (1.3)

where µ is the Haar measure and gα is in Cc(G) with support of gα iscontained in Vα. Then {fα}α∈I is a bounded approximate identity forL1(G).

Proposition 1.8. If f belongs to Lp(G), 1 ≤ p <∞ then

limα‖fα ∗ f − f‖p = 0 = lim

α‖f ∗ fα − f‖p

Proof. For any f in Cc(G) one can prove using Proposition 1.7 and thegeneral case follows by Lusin's theorem. �

6 V. MURUGANANDAM

Let us end our discussion about the group algebra by recalling someimportant properties of group algebras of commutative groups.We assume that G is commutative. A continuous group homomor-

phism from G into T, is called a character. The set of all charactersof G is denoted by G. It can be seen that G forms a locally com-pact abelian group under pointwise multiplication and compact - opentopology and is called as dual group of G. Let ∆(L1(G)) denote the setof all non-zero complex homomorphisms of L1(G).

Theorem 1.9. The map χ to τχ from G in to ∆(L1(G)) given by

τχ(f) =

∫G

f(x)χ(x)dx ∀f ∈ L1(G), (1.4)

de�nes a homeomorphism from G into ∆(L1(G)), with Gelfand topol-ogy.

De�nition 1.10. For every f ∈ L1(G) the Fourier transform of f is

de�ned to be a function on G given by

f(γ) =

∫G

f(x)〈 x, γ 〉 dx. (1.5)

Proposition 1.11. The Fourier transform is a injective homomor-phism from L1(G) to C0(G) and its range is dense in C0(G).

Theorem 1.12 (Fourier Inversion formula). Let G be a locally compact

abelian group. Then there exists a Haar measure dγ on G satisfyingthe following.

If f ∈ L1(G) and f ∈ L1(G), then

f(x) =

∫G

f(γ)γ(x) dγ a. e (1.6)

Theorem 1.13. [Plancherel theorem] Let G be a locally compact abeliangroup. If f ∈ L1(G) ∩ L2(G) then

‖f‖L2(G) = ‖f‖L2(G).

The Fourier transform extends to a unitary operator from L2(G) onto

L2(G).

For the proofs and more details we refer to Loomis[11] and the recentbook Folland [7]

7

1.3. Representation theory.

De�nition 1.14. A representation of G on a Hilbert space H is ahomomorphism π from G into GL(H), the group consisting of all in-vertible operators in BL(H), such that the map

x× → π(x)ξ

from G→ H is continuous for every ξ in H..

Here H is called the representation space. We say that the represen-tation π is unitary if π(x) is unitary for every x in G. That is,

〈 π(ξ)u, π(x)η 〉 = 〈 ξ, η 〉 ∀x ∈ G ∀ ξ, η ∈ H.

A representation (π,H) is said to be an irreducible representation ofG, if H does not have any non-trivial closed G-invariant subspace.A representation (π, H) is said to be cyclic with a cyclic vector ξ if

the linear span of {π(x)ξ : x ∈ G} is a dense subspace of H. Any two(unitary) representations (π1,H1) and (π2,H2) are said to be (unitarily)equivalent if there exists an invertible bounded (unitary) operator T :H1 → H2 such that

T ◦ π1(g) = π2(g) ◦ T, ∀ g ∈ G.

On any Hilbert space one can have trivial representation. That is,for every x in G, de�ne π(x) = IH. But this apart, there is a built-inrepresentation for every group, given as follows.For any x in G, if λ(x) denotes the operator on L2(G) given by

λ(x)f(y) = f(x−1y) ∀y ∈ G and ∀f ∈ L2(G),

then λ(x) is unitary as Haar measure is left invariant. Moreover, themap x→ λ(x)f de�nes a unitary representation of G and is known asleft regular representation of G.Notation Let G be the set of equivalence classes of irreducible unitaryrepresentation of G and let G be the set of equivalence classes of unitaryrepresentation of G.

Remark 1.15. 1. If H is a Hilbert space let H denote the conjugateHilbert space. If (π, H) is a representation de�ne π,H by π(x) = π(x).It is called conjugate representation.2. Let (πi,Hi), i = 1, 2 be two unitary representations of a groupG. Let H = H1 ⊕ H2 be the direct sum of Hilbert spaces. If we de�neπ(x)(ξ1, ξ2) = (π1(x)ξ1, π2(x)ξ2), then it forms a unitary representationcalled the direct sum of (π1,H1) and (π2,H2).

8 V. MURUGANANDAM

3. Let H be the tensor product H1⊗H2 of H1 and H2 obtained bycompleting the algebraic tensor product H1 ⊗ H2 by de�ning the innerproduct

〈 ξ1 ⊗ ξ2, η1 ⊗ η2 〉 = 〈 ξ1, η1 〉〈 ξ2, η2 〉.Recall that if T1 and T2 are bounded linear maps on H1 and H2 re-

spectively then there exists a bounded linear map T1 ⊗ T2 such thatT1 ⊗ T2(ξ1 ⊗ ξ2) = T1(x)ξ1 ⊗ T2(x)ξ2, and ‖T1 ⊗ T2‖ = ‖T1‖ ‖T2‖ .If π1 and π2 are two unitary representations of G then π1 ⊗ π2 on

H1⊗H2 given by

π1 ⊗ π2(x) = π1(x)⊗ π2(x)

de�nes a unitary representation on G. It is called tensor product of π1

and π2 and is denoted by (π1 ⊗ π2,H1⊗H2).

De�nition 1.16. Let

B(G) ={πξ, η

: ξ, η ∈ H, π ∈ G}.

By the above remark, we observe that B(G) is a subalgebra of Cb(G)closed under complex conjugation having identity. This space is goingto be an important object of study throughout this course.

De�nition 1.17. Let A be a Banach ?- algebra. Any ?-homomorphismπ of A into BL(H) for some Hilbert space H is called ?-representationof A.

Remark 1.18. Let us recall that if A be a Banach ?- algebra and B isa C?-algebra and if φ : A→ B is a ?-homomorphism then

‖φ(x)‖ ≤ ‖x‖for every x ∈ A. (see Takesakai [17] for a proof.) In particular any ?representation of L1(G) is norm decreasing.

Any ? representation π is said to be non-degenerate if the closureof the subspace spanned by [π(A)(H)] is equal to H. Similarly, we cande�ne �irreducibility� and �equivalence� among the ?-representations ofA.A ?-representation (π,H) of L1(G) is non-degenerate if and only if

the following holds. For any bounded approximate identity {fα} inL1(G), π(fα)→ I in strong operator topology.

Theorem 1.19. Suppose (π, H) is a unitary representation of G. Forevery f in L1(G) if we de�ne π(f) on H by

〈 π(f)(ξ) η 〉 =

∫G

f(x)〈 π(x)ξ, η 〉dx,

9

then f → π(f) de�nes a non-degenerate ?-representation of L1(G).Moreover the correspondence π → π is bijective from the equivalence

classes of all unitary representations of G and the equivalence classesof all non-degenerate ?-representations of L1(G).

Proof. Suppose that π is a unitary representation of G. If we de�ne πby

〈 π(f) =

∫G

f(x)π(x)dx, (1.7)

that is, for every ξ, η in H, if

〈 π(f)(ξ) η 〉 =

∫G

f(x)〈 π(x)ξ, η 〉dx,

then it can be easily seen to be a ?-homomorphism.Let {fα} be the bounded approximate identity given in (1.3). Fix

ξ in H and ε > 0 Since x → π(x)ξ is continuous at e there exists Vα0

such that, x ∈ Vα0 ⇒ ‖π(x)ξ − ξ‖ < ε. For every α > α0,

‖π(fα)ξ − ξ‖ =∥∥∥∫G

fα(x)π(x)ξ dx− ξ∥∥∥

≤∫Vα

|fα(x)| ‖π(x)ξ − ξ‖ dx

< ε.

∫Vα

|fα(x)| dx.

= ε.

Hence π(fα) → I in strong operator topology and so π|L1(G) is non-degenerate.Conversely suppose that ρ : L1(G) → BL(H) is a non-degenerate

?-representation of L1(G).Set K = [ρ(L1(G)(H)]. Then K is dense in H. De�ne

π(x)(ρ(f)ξ) = ρ(δx ∗ f)(ξ).

Since δx ∗ fα ∗ f → δx ∗ f, for all f ∈ L1(G), we have

π(x)(ρ(f)ξ) = limα‖ρ(δx ∗ fα)‖‖ρ(f)(ξ)‖

≤ ‖ρ(f)(ξ)‖.

Therefore by 1.18, π(x) is bounded on K. Hence π(x) gets extendedas an operator on H. It is easy to see that π(xy) = π(x) ◦ π(y). Since

10 V. MURUGANANDAM

‖ξ‖ = ‖π(x−1)π(x)(ξ)‖ ≤ ‖π(x)ξ‖ ≤ ‖ξ‖, π is unitary. Since for everyf, g in L1(G),

ρ(f) ◦ ρ(g) = ρ(f ∗ g)(ξ)

=

∫G

f(y)ρ(δy ∗ g)

=

∫G

f(y)π(y)(ρ(g))dy

= π(f)(ρ(g))

we have ρ(f) = π(f). �

Remark 1.20. For example λ, corresponding to λ is given by left con-volution operators, since

λ(f)(g)(x) =

∫G

λ(y)(g)(x)f(y)dy =

∫G

g(y−1x)f(y)dy = f ∗ g(x).

This representation is faithful in the sense that if λ(f) = 0 then f = 0in L1(G).

2. Positive definite functions

De�nition 2.1. A function φ : G→ C is said to be positive de�nite iffor all c1, c2, ..., cn ∈ C and x1, x2, ..., xn ∈ G,

n∑i,j=1

cicjφ(x−1j xi) ≥ 0.

Example 2.2. If (π,H) is any unitary representation of G then forany ξ H function φ = π

ξ,ξis positive de�nite since for any choice of xi

and ci as in the de�nitionn∑

i,j=1

cicjφ(x−1j xi) = 〈 η, η 〉

where η =∑

i ciπ(xi)(ξ).

Note that, taking n=1 and c1 = 1 in De�nition 2.1, we have φ(e) ≥ 0.

Proposition 2.3. Let φ be a positive de�nite function. Then

(1) φ(x−1) = φ(x)(2) |φ(x)| ≤ φ(e), ∀ x ∈ G.

Proof. By hypothesis, the matrix (φ(x−1i xj))1≤i,j≤n is positive semi de�-

nite. Consider A =

(φ(e) φ(x)φ(x−1) φ(e)

). Since A = A?, and det(A) ≥ 0,

the result follows.

11

�

Theorem 2.4. Let φ be a continuous function on G. Then the followingare equivalent

(1) φ is positive de�nite.(2) φ is bounded and 〈 φ, f ? ∗ f 〉 ≥ 0, ∀ f ∈ Cc(G).(3) 〈 φ, µ? ∗ µ 〉 ≥ 0, ∀ µ ∈M(G).

Proof. If µ is a measure with �nite support, that is if µ =n∑i=1

αiδxi ,

observe that

〈 φ, µ? ∗ µ 〉 =n∑

i,j=1

cicjφ(x−1j xi).

Assume that φ is positive de�nite. Then 〈 φ, µ? ∗ µ 〉 ≥ 0.Let f belong to Cc(G). Then there exists {µα} of measures with

�nite support such that µα converges to f(y)dy in the weak?-topology.Therefore, (1) implies (2).Let us prove (2) implies (3). Let us suppose that µ has compact

support. Then for any f in Cc(G) the function µ ∗ f belongs to Cc(G).If {fα} is a bounded approximate identity such that each fα belongs

to Cc(G), then µ ∗ fα belongs to Cc(G) and it converges to µ in theweak ?-topology. Therefore (3) is true in this case.If µ belongs to M(G), then there exists {µα} in M(G) with com-

pact support such that {µα} converges to µ in the weak ?- topology.Therefore (2) implies (3).It is trivial that (3) implies (1). �

Notation: Let P (G) denote the set of all continuous positive de�nitefunctions on G. If φ belongs to P (G) then observe that φ belongs toP (G).

Remark 2.5. In Example 2.2 we have seen that any matrix coe�-cient belonging to a unitary representation is positive de�nite. Now weshall show that these are the only positive de�nite functions. In otherwords, we show that if φ belongs to P (G), then there exists a cyclicrepresentation (π, H) with cyclic vector ξ such that φ(x) = π

ξ, ξ.

2.1. GNS construction.

Theorem 2.6. Let φ be any continuous positive de�nite function on G.Then there exists a cyclic representation (π, H) with the cyclic vectorξ such that φ(x) = 〈 πφ(x)u, u 〉 locally almost everywhere.

12 V. MURUGANANDAM

Proof. De�ne 〈 ·, · 〉φ on L1(G) by

〈 f, g 〉φ = 〈 g? ∗ f 〉 =

∫G

φ(x−1y) ¯g(x)f(y)dxdy.

It is easy to see that it de�nes a sesquilinear form on L1(G). If N ={f ∈ L1(G) : 〈 f, f 〉φ = 0} , one can see by Cauchy Schwarz inequalityone can see that N = {f ∈ L1(G) : 〈 f, g 〉φ = 0∀g ∈ L1(G)} . There-fore it forms a closed subspace of L1(G).Moreover, since

〈 xf,x g 〉φ = 〈 f, g 〉φ (2.1)

we see that N is invariant under left translation.Let H0 denote the quotient space L1(G)/N . Complete it to get a

Hilbert space H. We shall de�ne a representation π on H as follows. Iff belongs to L1(G)/N take

π(x)(f) = x−1f = x−1 f

Then π extends to a unitary representation of G.Let us show that (π, H) is cyclic. If {fα} is a bounded approximate

identity of L1(G) then take a subnet if necessary to conclude that fαconverges to a vector ξ weakly in H. Then

〈 f , ξ 〉φ = limα〈 f , fα 〉φ = lim

α〈 f ?α ∗ f, φ 〉 =

∫G

f(x)φ(x)dx. (2.2)

Since 〈 g, f 〉φ =∫G

∫Gφ(x−1y) ¯f(x)g(y)dxdy and∫

G

φ(x−1y)g(y)dy =

∫G

φ(y)g(xy)dy = 〈 π(x−1)g, ξ 〉φ = 〈 g, π(x)ξ 〉

we see by (2.2) that

〈 g, f 〉φ =

∫G

¯f(x)〈 g, π(x)ξ 〉

= 〈 g, π(f)(ξ) 〉φ. (2.3)

Hence [π(L1(G))ξ] is total in H and so the representation is cyclic.Finally,

〈 ξ, f 〉 = limα〈 fαf 〉,= lim

α〈 fα, π(f)ξ 〉 = 〈 ξ, π(f)(ξ) 〉

Therefore, by (2.2),∫Gf(x)φ(x)dx = 〈 π(f)(ξ), ξ 〉 �

Remark 2.7. Using the preceding theorem, we conclude that the vectorspace B(G) de�ned in 1.16 is in fact linear span of continuous positivede�nite functions.

13

3. C? algebras of groups

De�nition 3.1. An Banach ?- algebra A is said to be C?-algebra if theinvolution of A satis�es the additional condition

‖x?x‖ = ‖x‖2, x ∈ A.

Remark 1. Let X be a compact Hausdor� space. The space C(X) ofcontinuous complex valued functions on X is a unital Banach algebrawith the uniform norm. The map f → f is an involution that makesC(X) into a C?-algebra. Similarly ifX is a locally compact noncompactHausdor� space then C0(X) consists of continuous functions whichvanish at in�nity forms a C?-algebra without identity.2. L∞(X, dµ) for any measure µ, is a C?-algebra.3. Let H be a Hilbert space. Then the unital Banach algebra BL(H) isa C? algebra with the operator norm and the involution given by themap T → T ?. In general any norm closed ?-subalgebra of BL(H) is aC?-algebra.

Theorem 3.2. If ‖·‖′ is de�ned on L1(G) by

‖f‖′ = sup {‖π(f)‖ : π is any non-degenerate ?-representation} (3.1)

then it de�nes a norm on L1(G). Moreover, the completion of L1(G)with respect to this norm is a C?-algebra.

Proof. Clearly,

‖λf + g‖′ ≤ |λ| ‖f‖′ + ‖g‖′

Suppose that ‖f‖′ = 0. Then λ(f) = 0 where λ : L1(G) → BL(L2(G))is the left regular representation. As λ is a faithful representation, wehave f = 0. Therefore ‖.‖′ is indeed a norm. �

De�nition 3.3. The C?-algebra obtained above is called full C?-algebraof G and is denoted by C?(G).

The following theorem gives yet another way to realize C?(G). Letus �rst recall

Theorem 3.4. A C?-algebra has su�ciently many irreducible repre-sentations to separate points of A.That is, for every x ∈ A, x 6= 0 there exists an irreducible represen-

tation π of A such that π(x) 6= 0.

Theorem 3.5. let A denote the C?(G)-algebra obtained by completing

L1(G) with ‖ · ‖′′ where ‖f‖′′ = sup{‖π(f)‖ : π ∈ G

}. Then C?(G)

and A are isometrically ?isomorphic

14 V. MURUGANANDAM

Proof. De�ne Ψ : (L1(G), ‖.‖′)→ (A, ‖.‖′′) . Then ‖Ψ(f)‖ ≤ ‖f‖′. So Ψextended into a ?-homomorphism from C?(G) into A.We claim that Ψ

is injective. Suppose Ψ(f) = 0,∀ f ∈ C?(G). Then π(f) = 0, ∀ π ∈ G.By the preceding Gelfand-Raikov Theorem, f = 0. Therefore Ψ isinjective. Since an injective ?-homomorphism between C?-algebras isan isometry, we have

‖Ψ(f)‖ = ‖f‖, ∀ f ∈ C?(G).

Therefore, Ψ(C?(G)) is closed in A. Thus Ψ(C?(G)) = A. Hence A isC? -completion of (L1(G), ‖.‖′′)

�

Proposition 3.6. Suppose that the group G is abelian. Then the fullC?-algebra of G is identi�ed with C0(G).

Proof. If (π,H) is a unitary irreducible representation of G then bySchur's lemma it is given by one-dimensional representation. That is,there exists a character χπ such that π(x)ξ = χπ(x), for every x ∈ G.π going to χπ identi�es G with the dual group given in Section 1.Now

π(f) =

∫G

f(x)π(x) dx. =

∫G

f(x)χπ(x) dx. = f(χπ),

where χπ = χπ(x−1). Therefore,

‖f‖′ = supπ∈ G‖π(f)‖ = sup

χ∈ G‖f(χπ)‖ = ‖f‖∞, since χ ∈ G.

Since C?(G) is the completion of (L1(G), ‖.‖′) and ∆(L1(G)) = G,

we have C?(G) is the completion of {f ∈ C0(G) : ‖.‖∞ − norm}. Since{f : f ∈ Cc(G)} is dense in L1(G), we have C?(G) = C0(G). �

Remarks 3.7. 1. If ρ is any ?-representation of L1(G), then ρ getsextension to a ?-representation of C?(G).2. If ρ1 and ρ2 are two non-degenerate representations of L1(G) thenthey are equivalent if and only if their extensions to C?(G) are equiva-lent.3. Summarizing by we observe by Theorem 1.19 and the preceding the-orem that there is a bijective correspondence between the unitary rep-resentations of G and non-degenerate ?-representations of C?(G) suchthat irreducible ones go into irreducible ones. Moreover this identi�ca-tion respects equivalence relation among the representations.

15

From the proof of Theorem 3.2 we observe that instead of taking allunitary representations in the equation (3.1), if we take the left regularrepresentation alone, we get another C?-algebra.

De�nition 3.8. The closure of ?-subalgebra λ(L1(G)) in BL(L2(G))is a C?-algebra and is called reduced C?-algebra of G.We denote this C?-algebra by C?

λ(G).

We recall that if G is abelian then∣∣∣f(γ)

∣∣∣ ≤ ‖λ(f)‖ for every f in

L1(G) and for every γ in G. So ‖f‖′ = ‖λ(f)‖ for every f in L1(G), sothat C?(G) ∼= C?

λ(G).Similarly if G is compact, by Peter-Weyl theory we observe that‖π(f)‖ ≤ ‖λ(f)‖ for every π in G and so C?(G) ∼= C?

λ(G).The question that for which groups, these two C?-algebras are iso-

metrically ?-isomorphic gives rise to a class of groups called amenablegroups, which are to be discussed below.

3.1. Weak containment:

De�nition 3.9. Suppose that G is a locally compact group. Let Σ ⊆G, π ∈ G. We say that π is said to be weakly contained in Σ if thecorresponding ?-representation π of C?(G) is weakly contained in cor-responding Σ. We denote it by π � Σ.

We recall that if π is a representation of A and Σ is a collection ofrepresentations of A then π is weakly contained in Σ denoted by π � Σif any of the equivalent conditions in the following theorem is satis�ed.

Theorem 3.10. (1) ker(π) ⊇⋂ρ∈Σ

ker(ρ)

(2) ‖π(x)‖ ≤ sup{‖ρ(x)‖ : ρ ∈ G

}.

(3) For every ξ ∈ Hπ, there exists a net {φα} consisting of thematrix coe�cients belonging the representations in Σ such that{φα} converges to πξ, ξ in the weak? topology. In fact Everypositive form πξ,ξ associated to π is weak? limit of linear sum ofpositive linear form associated to Σ.

(4) Every state of A associated with π is a weakstar limit of stateswhich are sums of positive forms associated with Σ.

See Theorem 3.4.4 of Diximier [4]. For more details regarding theweak containment among the representations of a C?-algebra we referto Section 3.4 of Diximier [4].If we take Σ = {λ} then we observe by the preceding discussion

and Theorem 3.5, that C?(G) and C?λ(G) are isometrically isomorphic

16 V. MURUGANANDAM

if and only if every irreducible unitary representation of G is weaklycontained in λ.The following theorem is useful in understanding the weak contain-

ment among the representations of G.

Theorem 3.11. Let P1(G) = {ψ ∈ P (G) : ‖ψ‖ = ψ(1) = 1} . On P1(G)the weak? topology and the topology of uniformly convergence on com-pact sets are equivalent.

See Theorem 13.5.2. of Diximer [4].

Theorem 3.12. Let G be a locally compact group and π belong to Gand Σ ⊆ G. Then the following are equivalent:

(1) π � Σ.(2) Every positive de�nite function ψ associated to π is limit of sum

of positive de�nite functions associated to Σ with respect to thetopology of uniformly convergence on compact sets.

If π is further assumed to be irreducible, then the above if equivalentto the following.3. If πξ,ξ for some ξ in H, is the limit of sum of positive de�nite

functions associated to Σ with respect to the topology of uniformly con-vergence on compact sets.

Proof. The proof follows by Theorem 3.10, Theorem 3.11.�

3.2. Fourier and Fourier Stieltjes algebra.

Theorem 3.13. Let G be a locally compact group. Let C?(G) denotethe C?-algebra of G. The Banach space dual [C?(G)]? of C?(G) is givenby

{πξ,η : ξ, η ∈ Hπ, π is a unitary representation of G} . (3.2)

We shall �rst prove the following lemma.

Lemma 3.14. If φ is a positive linear form L1(G), then φ gets extendedto a positive linear form φ′ on C?(G). The map φ→ φ′ is bijective and‖φ‖ = ‖φ′‖.

Proof. Let τ : L1(G) → C?(G) denote the imbedding. As every pos-itive form on a Banach ?-algebra with bounded approximate identityis continuous (See [1, p396]). Therefore any positive form on L1(G)gets extended to φ′ on C?(G) such that, φ = φ′ ◦ τ. Let x ∈ C?(G).We need to see that ‖φ‖ = ‖φ′‖. If f ∈ L1(G) then |φ(f)| = |φ′(f)| ≤

17

|φ′(e)|1/2‖f ? ∗ f‖1/2C?(G) ≤ ‖φ′‖‖f‖1. Therefore ‖φ‖ ≤ ‖φ′‖. Let x ∈ A.

Then,

|φ′(x)| = |φ(x)|≤ ‖φ‖1/2(φ(x?x))1/2.

≤ ‖φ‖1/2‖φ′‖1/2‖x?x‖C?(G)

≤ ‖φ‖1/2‖φ′‖1/2‖x‖2.

e Therefore, ‖φ′‖ ≤ ‖φ‖1/2‖φ′‖1/2 ⇒ ‖φ′‖1/2 ≤ ‖φ‖1/2 ⇒ ‖φ′‖ ≤ ‖φ‖.Hence ‖φ‖ = ‖φ′‖ as claimed.

�

Let us prove the theorem.

Proof. Let us recall Jordan Decomposition Theorem for C?-algebras.(see Theeorem (3.2.5) of Pederson [13]). For each hermitean functionalφ on C?-algebra A, there exist positive elements φ+ and φ− such thatφ = φ+−φ− and ‖φ‖ = ‖φ+‖+‖φ−‖. By GNS construction theorem forC?-algebras we see that if φ is any positive linear form, then there existsa unique (up to unitary equivalence) cyclic ?-representation (π,H, ξ)with the cyclic vector ξ, satisfying

φ(x) = 〈 π(x)(ξ), ξ 〉 ∀ x ∈ A.Therefore, we observe that any element in the dual of a C?-algebra isa linear combination of positive linear forms. By the preceding lemma,and Remark 3.7 the result follows. �

Remark 3.15. The vector space B(G) is identi�ed with the Banachspace dual [C?(G)]? of the C?-algebra of G and is called Fourier-Stieltjesalgebra of G.

In particular, if G is abelian then B(G) w [C0(G)]? = M(G). In factthe identi�cation is given by the inverse Fourier-Stieltjes transform.That is

B(G) = {φ = µ : for some µ ∈M(G)}, with ‖φ‖B(G) = ‖µ‖ ,where the inverse Fourier-Stieltjes transform is given by

µ(γ) =

∫G

γ(x)dµ(x).

But then this is precisely Fourier stieltjes algebra on G. Refer the clas-sical book by Rudin [15] for more details on harmonic analysis overabelian groups. The result that it forms a Banach algebra when G is

18 V. MURUGANANDAM

abelian was extended to all non-abelian groups by Eymard [6]. Thefollowing theorem is due to Eymard.

Theorem 3.16. B(G) forms a Banach algebra with unity under point-wise product.

Proof. We have already seen that B(G) forms an algebra with unityunder pointwise product. By the preceding remark it forms a Banachspace. In order to show that the norm satis�es Banach algebra condi-tion we need the following fact due to Eymard.If φ belongs to B(G) then

‖φ‖ = inf{‖ξ‖‖η‖ : φ = πξ,η}, (3.3)

where the in�rmum is taken over all πξ,η such that φ = πξ,η. (In factthe minimum is attained.)Using the above equation it is easy to show that ‖φ · ψ‖ ≤ ‖φ‖ ‖ψ‖ ,

for all φ, ψ in B(G). �

De�nition 3.17. Let G be locally compact group. The closure of theideal B(G)∩Cc(G) in B(G) is called the Fourier algebra and is denotedby A(G).

When G is abelian, recall that Fourier algebra

A(G) = {f : f ∈ L1(G)}where the f denote the inverse Fourier transform of f and

∥∥f∥∥A(G)

=

‖f‖L1(G) . It is one of the classical results of abelian harmonic analysis

that whose proof can be found in Rudin [15].Again, when G is abelian, one can arrive at the following character-

ization of A(G) using Plancherel theorem.

A(G) ={f ∗ g, f, g ∈ L2(G)

},

where g(y) = g(y−1). This result was extended to all locally compactgroups by Eymard.

Theorem 3.18 (Eymand). Let G be a locally compact group. Then

A(G) ={λf,g : f, g ∈ L2(G)

},

where λf,g is the matrix coe�cient associated to the left regular repre-sentation λ of G.

1. A(G) has identity if and only if A(G) = B(G) since A(G) is atwo sided ideal in B(G). Equivalently, A(G) = B(G) if and only if Gis compact.2. A(G) is a commutative regular, semi simple Tauberian Banach

algebra.

19

3. The canonical embedding of G in ∆(A(G)) namely, x into τx givenby τx(φ) = φ(x) is a bijective homeomorphism.For the proofs of all these results we refer to Eymard [6].

4. Amenable groups

De�nition 4.1. A linear map m : L∞(G)→ C is said to be a mean onL∞(G), if m(f) ≥ 0 for all f ≥ 0 in L∞(G) and m(1) = 1. Moreover,a mean is said to be a left invariant mean if

m(xf) = m(f) ∀f ∈ L∞(G),∀x ∈ G.

De�nition 4.2. A locally compact group G is said to be amenable ifL∞(G) has a left invariant mean.

Amenable groups were �rst introduced by John von Neumann in1929 in his study of Banach-Tarski paradox. (See a fairly recent bookby Runde[16] for a discussion on Banach-Tarski paradox.) But it wasM.M. Day [2], who baptized the name. As we are going to see below,amenable groups form a vast collection of groups, which include forinstance abelian groups, solvable groups and compact groups.Any compact group is amenable. In fact the normalized Haar mea-

sure is the required left invariant mean. That is, if 〈 f,m 〉 =∫Gf(x)dx

then m is easily seen to be a left invariant mean.We shall use Markov- Kakutani �xed point theorem1 to show that

any abelian group is amenable.IfK denotes the set of all means on L∞(G), then it forms a nonempty

convex set. Moreover, it is a weak?-compact set as it is a subset of theunit ball of the dual of L∞(G).For all x ∈ G, de�ne ρ(x) : L∞(G)? → L∞(G)? by

〈 f, ρ(x)(F ) 〉 = 〈 x−1f, F 〉 ∀F ∈ L∞(G)?.

Then it is easy to see that ρ(x) is continuous, linear on L∞(G)? andleaves K invariant. Since ρ(xy) = ρ(x) ◦ ρ(y), for all x, y in G and Gis abelian, {ρ(x)}x∈G forms a commuting family. If m is a �xed point,then m is left invariant.The following theorem is useful. We refer to Runde [16] for the proof.

Theorem 4.3. Let G be a locally compact group. Then the followingare equivalent,

(1) G is amenable.

1Markov- Kakutani �xed point theorem: If K is a compact convex subset ofa topological vector space and if F is a commuting family of continuous linearmappings which map K into itself, then there exists a point p in K such thatT (p) = p for all T in F . (See Dunford and Scwartz[5, Theorem V.10.6] for a proof.)

20 V. MURUGANANDAM

(2) Cu(G) has a left invariant mean.(3) Cb(G) has a left invariant mean.

Corollary 4.4. Let Gd denote G with discrete topology. Then if Gd isamenable then G is amenable.

Proof. Suppose Gd is amenable. Then l∞(G) has left invariant mean,say m. Observe that Cb(G) ⊆ l∞(G). Then m|Cb(G) is left invariantmean on Cb(G). Therefore G is amenable. �

The following properties of amenable groups are useful to generatemore examples of amenable groups and nonamenable groups.

Lemma 4.5. Let H be a locally compact group, and let φ : G → Hbe a continuous, open homomorphism with φ(G) is dense in H. If G isamenable then H is amenable.

Proof. Let m denote a left invariant mean on Cb(G).De�ne a continuous homomorphism φ? : Cb(H)→ Cb(G) by

φ?(f)(g) = f(φ(g)), g ∈ G.

If we de�ne m on Cb(H) by

〈 f, m 〉 = 〈 φ?(f),m 〉,

then it is easy to see that m is left invariant mean on Cb(H).�

Corollary 4.6. Let G be amenable, and let N be a closed normalsubgroup of G. Then G/N is amenable.

Proposition 4.7. Let H be a closed subgroup of G. Then there is aBruhat function for H. That is, there exists a function β : G → Cassociated to H, called Bruhat function satisfying the following:

(1) β - is continuous and positive.(2) For all compact set K, support of (β|KH) is compact.(3) For all g ∈ G,

∫H

β(gh) dh = 1.

Theorem 4.8. Let H be a closed subgroup of G. If G is amenable,then H is also amenable.

Proof. If β denotes the Bruhat function associated to H, de�ne T :Cb(H)→ l∞(G) by

Tφ(g) =

∫G

β(g−1h)φ(h) dmH(h), g ∈ G.

21

We show that T is a contraction from Cb(H) into Cb(G) mappingthe constant function 1H into 1G.Let φ belongs to Cb(G). Then we �rst show that T (φ) is a continuous

function on G.Fix g0 ∈ G. If V is a compact neighbourhood V of g0, then by con-

dition (2) of the preceding proposition, β|V H is uniformly continuous.There exists a neighbourhood W of e such that

| β|V H(g)− β|V H(h)| < ε.

for all h in gW. In particular if g ∈ g0W, then g ∈ V H and∣∣β(g−1h)− β(g−10 h)

∣∣ < ε

mH(support of (β|V H))‖φ‖∞.

Now it can be seen that | Tφ(g)−Tφ(g0) |≤ ε. If m is a left invariantmean on Cb(G), de�ne m on Cb(H) by 〈 φ, m 〉 = 〈 Tφ,m 〉. It is routineto check that m is left invariant mean.

�

Theorem 4.9. Let N be a closed normal subgroup such that both Nand G/N are amenable. Then G is amenable.

Proof. LetmN be the left invariant mean on Cb(N).De�ne T : Cb(G)→Cb(G) by

Tφ(g) = 〈 ( g−1φ)|N ,mN 〉.Then T (φ) is continuous, and ‖Tφ‖∞ ≤ ‖φ‖∞, mapping the constantfunction 1G into itself. Use the left invariance of mN to conclude thatTφ(gh) = Tφ(g), for all g in G, and h in H.

Therefore Tφ induces a function say T φ on G/N by

T φ(g) = Tφ(g).

Now if m is a left invariant mean on Cb(G/N) then de�ne m on Cb(G)by

〈 φ,m 〉 = 〈 T φ, m 〉.One can verify that m de�nes a left invariant mean on Cb(G).

�

For a detailed proof of these theorems, we refer to Greenleaf [8] andRunde [16].

Remark 4.10. Any solvable group is solvable.Since any abelian group is amenable, we infer by preceding theorem,

that Gd is amenable and so by Corollary 4.4, G is amenable.

Now let us give some examples which are not amenable.

22 V. MURUGANANDAM

Theorem 4.11 (Von-Neumann). F2 is not amenable.

Proof. Suppose on the contrary that there exists a left invariant meanm on L∞(G).Let W (x) = {w ∈ F2 : w starts with x} . Then F2 = {e} ∪W (a) ∪

W (a−1) ∪W (b) ∪W (b−1) and the union is disjoint.If w belongs to F2 \W (a), then a−1w belongs to W (a−1). That is,

w belongs to aW (a−1). Therefore, F2 = W (a) ∪ aW (a−1). SimilarlyF2 = W (b) ∪ bW (b−1).Now, if χE denotes the characteristic function of a set E, then

1 = m(1)

= m({e}) +m(χW (a)) +m(χW (a−1)) +m(χW (b)) +m(χW (b−1))

≥ m(χW (a)) +m(χaW (a−1)) +m(χW (b)) +m(χbW (b−1))

= m(χW (a)∪aW (a−1)) +m(χW (b)∪bW (b−1))

= m(χF2) +m(χF2) = 2.

So, m cannot be a left invariant mean. That is, F2 is not amenable. �

Using Theorem 4.8 and the preceding theorem, we conclude that

Corollary 4.12. Any locally compact group that contains F2 as a closedsubgroup, is not amenable.

For instance, SL(2,R) is not amenable since the the closed subgroup

generated by elements a =

(1 20 1

)and b =

(1 02 1

)is isomorphic

with F2.Similarly we see that SL(n,R) is not amenable for all n > 2, be-

cause SL(n,R) contains SL(2,R) as closed subgroup. With the samereason GL(n,R) is seen to be not amenable. In fact, any connectednoncompact semisimple Lie group is not amenable.From the above we learn that a locally compact group containing

F2 is not amenable. It is worthwhile to know whether there exist dis-crete groups which are non-amenable groups and do not contain F2.Wewish to point out that the existence of such groups remained a funda-mental open problem (sometimes called von Neumann problem) untilOlshanskii[12] established such groups in 1980. There exists anotherclass of in�nite discrete groups (à la Gromov) with Kazdhan propertyT, (much stronger than non-amenability), yet not having any free sub-group.

23

5. Some characterizations of Amenable groups

Theorem 5.1. Let G be a locally compact group. Then the followingare equivalent:

(1) G is amenable.(2) There exists a net {gi} in S(G) such that {h ∗ gi − gi} → 0 in

the weak? topology of (L∞(G))?, ∀ h ∈ S(G).(3) There exists a net {gi} in S(G) such that, lim

i‖h ∗ gi − gi‖1 =

0, ∀ h ∈ S(G).(4) For given ε > 0, for every compact set K, there exists g ∈ S(G)

such that ‖λ(x)g − g‖1 < ε, ∀ x ∈ K.

Remark 5.2. Condition (iv) of the preceding theorem is called P1-property. More generally,

De�nition 5.3. We say that a locally compact group G is said to havethe property Pp, 1 ≤ p <∞ if the following holds.For given ε > 0, for every compact set K, there exists g ∈ Lp(G)

such that g is non-negative and ‖g‖p = 1 satisfying

‖λ(x)g − g‖p < ε, ∀ x ∈ K.

Theorem 5.4. Let G be a locally compact group. Then G has propertyP1 if and only if G has the property Pp , ∀ p such that 1 ≤ p <∞.

See Reiter [14] and Runde [16] for the proofs of the theorems citedabove.

Theorem 5.5. G is amenable if and only if there is a net {fi} in the

unit sphere of L2(G) such that {fi ∗ fi} converges to 1 uniformly oncompact subsets of G.

See Pederson [13, Proposition 7.3.8].In fact we can say something more.

Theorem 5.6. Let G be a locally compact group. Then G has propertyP2 if and only if there exists {fi} ⊆ Cc(G) such that ‖fi‖2 = 1 and

fi ∗ fi → 1 uniformly on compact subsets of G.

Proof. Let K be a compact subset of G and let ε > 0 be given. Choosef ∈ Cc(G) be such that ‖f‖2 = 1 and |(f ∗ f)(x) − 1| < ε

2, ∀ x ∈ K.

24 V. MURUGANANDAM

Then we have (f ∗ f)(e) = 1. Furthermore,∥∥λ(x)f − f∥∥2

2= 〈 λ(x)f − f, λ(x)f − f 〉

= 〈 λ(x)f, λ(x)f 〉+ 〈 f, f 〉 − 2Re(〈 λ(x)f, f 〉

)= 2− 2Re(f ∗ f(x))

= 2Re(1− (f ∗ f)(x)

)≤ 2

∣∣1− (f ∗ f)(x)∣∣ < ε.

If we take g = |f | then,∥∥λ(x)g − g∥∥2

2=∥∥λ(x)f − f

∥∥2

2< ε.

Therefore G has the property P2.Conversely, suppose G has the property P2. Then for given ε > 0, for

any compact set K, there exists f ∈ L2(G) such that ‖λ(x)f − f‖2 <ε2, f ≥ 0 and ‖f‖2 = 1, ∀ x ∈ K. Since Cc(G) is dense in L2(G), givenε > 0 there exists g ∈ Cc(G) with ‖g‖2 = 1 such that ‖g − f‖ < ε

4.

Therefore, for any x ∈ K, we have|g ∗ g(x)− 1| ≤ |g ∗ g(x)− f ∗ f(x)|+ |f ∗ f(x)− 1|

=∣∣g ∗ (g − f)(x) + (g − f) ∗ f(x)|+ |f ∗ f(x)− 1|.

Therefore

|g ∗ g(x)− 1| ≤ ‖g − f‖(‖g‖+ ‖f‖

)+ ‖λ(x)f − f‖ ≤ ε.

We deduce that there exists {g} in Cc(G) with ‖g‖2 = 1 such thatg ∗ g → 1 uniformly on compact subsets of G. Hence the theorem. �

Let G be a locally compact group. Then G is amenable if and only ifthere exists {fi} ⊆ Cc(G) such that ‖fi‖2 = 1 and fi∗ fi → 1 uniformlyon compact subsets of G. Let us recall an all time important theoremin the representation theory of groups, namely, Godement's theorem.

Theorem 5.7 (Godement). Let φ be any square-integrable continuouspositive de�nite function on G. Then there exists a square-integrable

function ψ such that φ = ψ ∗ ψ.

Refer Dixmier [4, Theorem 13.8.6] for a proof and more details.

Theorem 5.8 (Hulanicki). Let G be a locally compact group. Thenthe following are equivalent.

(1) G is amenable.(2) Every irreducible unitary representation of G is weakly con-

tained in λ.(3) The trivial representation G is weakly contained in λ.

25

Proof. We shall �rst show that (1) and (2) are equivalent. Suppose Gis amenable. Then there exists {fi} in Cc(G) such that ‖fi‖ = 1 and

fi ∗ fi → 1 in topology of uniform convergence on compact sets. Thatis, if φi = fi ∗ fi then φi → 1 in topology of uniform convergence oncompact sets. Hence 1 � λ, by (3) of Theorem 3.12.Conversely, suppose that if 1 � λ. Then by Theorem 3.12 and by

Theorem 5.5 we observe that G is amenable.Now we shall show that (2) and (3) are equivalent. Of course one

has to show only that (3) implies (2). Let the trivial representationis weakly contained in λ. By the above paragraph G is amenable.Therefore by the preceding remark, there exists a net of positive def-inite functions {ψi} in Cc(G), such that {ψi} → 1 in topology of uni-

formly convergence on compact sets. Let π ∈ G and φ = πξ,ξ. Then{φψi} → φ and φψi belongs to Cc(G). By Godement's theorem, thereexists gi ∈ L2(G) such that φψi = gi ∗ g. Thus gi ∗ gi → φ in topology ofuniform convergence on compact sets. Hence by Theorem 3.12, againwe have that π is weakly contained in λ.

�

The preceding theorem is due to Hulaniki [9]. Finally,

Corollary 5.9. A group G is amenable if and only if C?(G) ∼= C?λ(G).

Lemma 5.10. If {uα} is a bounded approximate identity in A(G).Then {uα} → 1 uniformly on compact sets.

Proof. Recall that for every φ = πξ,ξ in B(G),

|φ(x)| = |〈 π(x)ξ, η 〉| ≤ ‖ξ‖ ‖η‖.Therefore

‖φ‖∞ ≤ ‖φ‖.Let K be a compact set. Since A(G) is regular, there exists φ ∈ A(G)such that φ|K = 1. For every x ∈ K,

|uα(x)− 1| = |(uα.φ(x))(x)− φ(x)| ≤ ‖uα.φ− φ‖A(G).

Hence {uα} → 1 uniformly on compact sets. �

Theorem 5.11 (Leptin). Let G be a locally compact group. If G isamenable if and only if A(G) has a bounded approximate identity.

Proof. Let G be amenable. Then there exists {fi} ⊆ Cc(G) such that

‖fi‖2 = 1 and fi∗ fi → 1 uniformly on compact subsets of G. Denote ψiby fi ∗ fi.We claim that ψi is a bounded approximate identity in A(G).Since Cc(G) ∩ P (G) spans a dense subsets of A(G) and ‖ψi‖A(G) ≤

26 V. MURUGANANDAM

‖fi‖2 ‖fi‖2 = 1, it is su�cient to show that limi‖ψiφ−φ‖A(G) = 0, ∀ φ ∈

Cc(G)∩P (G). Let φ ∈ Cc(G)∩P (G). Take χi = ψiφ. Then χi ∈ P (G).For f ∈ Cc(G), we let ρ(f) denote the bounded operator on L2(G)given by convolution

ρ(f)g = g ∗ f, f ∈ L2(G).

By Godement's theorem, ρ(χi), ρ(φ) are positive and there exists hi, h ∈L2(G) such that

χi = hi ∗ hi, φ = h ∗ hand

ρ(χi)1/2(f) = f ∗ hi , ρ(φ)1/2(f) = f ∗ h, ∀ f ∈ Cc(G).

We shall show that limi‖hi−h‖2 = 0. Let S = supp(φ). Then supp(χi) ⊆

S. Since ψi → 1 uniformly on S, we have

limi‖χi − φ‖∞ = 0. (5.1)

For any f ∈ Cc(G), let fy(x) = f(xy−1), x ∈ G. Then clearly, ‖fy‖2 =∆(y)1/2‖f‖2. Therefore for any f ∈ Cc(G), we have∥∥ρ(χi)− ρ(φ)

∥∥ =∥∥f ∗ (χi − φ)

∥∥2

=∥∥∥∫G

(χi(y)− φ(y)

)∆(y)−1fy dy

∥∥∥2

≤

(∫G

∣∣χi(y)− φ(y)∣∣ ∆(y)−1/2 dy

)‖f‖2

≤

(∫S

∆(y)−1/2 dy

) ∥∥χi − φ∥∥∞ ‖f‖2.

Therefore, by (5.1), we have limi

∥∥ρ(χi) − ρ(φ)∥∥ = 0 and the num-

ber c = sup ‖ρ(χi)‖ is �nite. By approximating the function√t with

polynomials uniformly on the interval [0, c] we see that

limi

∥∥ρ(χi)1/2 − ρ(φ)1/2

∥∥ = 0.

Thus for every f ∈ Cc(G),

limi

∥∥f ∗ (hi − h)∥∥

2= lim

i

∥∥(ρ(χi)1/2 − ρ(φ)1/2

)(f)∥∥

2= 0.

Since f ?(x) = ∆(x)−1f(x−1), x ∈ G, we havelimi〈 hi − h, f ∗ g 〉 = lim

i〈 f ? ∗ (hi − h), g 〉 = 0, ∀ f, g ∈ Cc(G).

27

Moreover, ∥∥hi∥∥2

2=

∫G

|hi(x)|2dx =

∫G

hi(x)hi(x)dx

=

∫G

hi(x)hi(x−1)dx = (hi ∗ hi)(e).

Since Cc(G) ∗Cc(G) is dense in L2(G) and sup ‖hi‖22 = supχi(e) <∞,

we havelimi〈 hi, f 〉 = 〈 h, f 〉, f ∈ L2(G).

Also limi‖hi‖2 = lim

iχi(e) = φ(e) = ‖h‖2. Therefore,

limi‖hi − h‖2

2 = 2‖h‖22 − 2 lim

iRe〈 hi, h 〉 = 0.

Now, ∥∥ψiφ− φ∥∥A(G)= ‖χi − φ‖A(G)

=∥∥hi ∗ hi − h ∗ h∥∥A(G)

=∥∥hi ∗ (hi − h) + (hi − h) ∗ h‖A(G)

≤ ‖hi‖2 ‖hi − h‖2 + ‖hi − h‖2 ‖h‖2

= ‖hi − h‖2

(‖hi‖2 + ‖h‖2

).

Therefore, limi

∥∥ψiφ − φ∥∥A(G)

= 0. Similarly we can get limi

∥∥φψi −φ∥∥A(G)

= 0. Hence ψi is a bounded approximate identity in A(G) as

claimed.Conversely, suppose that {uα} is a bounded approximate identity. Weclaim that A(G) is weak? dense in B(G). Let φ ∈ B(G). Consider{φ.uα} is a bounded net in B(G). Any bounded set in B(G) is weak?

compact. By Lemma 5.10, uα → 1 uniformly on compact sets. There-fore φ.uα → φ. By Theorem 3.11, φ.uα → φ in weak? topology. SinceA(G)is dense in B(G), uα ∈ A(G). Therefore A(G) is weak? dense in B(G).Consider

A(G)+ = {φ ∈ A(G) : φ is positive de�nite}.We show that A(G)+ is weak? dense in B(G)+ = P (G). Since A(G)+

is a convex set, there exists T ∈ C?(G) such that

〈 T, u 〉 = 0, ∀ u ∈ A(G)+.

As A(G) is weak? dense in B(G), T = 0.Since (Cc(G)∩A(G))+ is dense in A(G)+, (Cc(G)∩A(G))+ is weak?

dense in B(G)+. As 1 ∈ B(G), there exists φi ∈ Cc(G) ∩ A(G) and

28 V. MURUGANANDAM

bounded such that φi → 1 in weak? topology. Therefore φi → 1 intopology of uniform convergence on compact sets. By Godement'stheorem, there exists {fi} in L2(G) such that φi = fi ∗ fi. Hence G isamenable. This completes the proof of the Theorem 5.11.

�

The preceding theorem is due to Leptin [10]. But the proof givenhere is adopted from what is given in Pederson [13]. See also Appendixof De Canniere and Haagerup [3].

References

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3. J. DeCannière and U. Haagerup, Multipliers of the Fourier algebra of some

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group, Russian Math. Surveys 35 (1980), 180�181.13. Gert K. Pederson, C?-algebras and their automorphism groups, London Math.

Society Monographs, no. 14, Academic Press, London-New york, 1979.14. H. Reiter and J. D. Stegeman, Classical harmonic analysis and locally compact

groups, second ed., Oxford University Press, Oxford New York, 2000.15. W. Rudin, Fourier analysis on groups, Interscience, New York - London, 1962.16. V. Runde, Lectures on amenability, Lecture Notes in Mathematics, vol. 1774,

Springer, Berlin, Heidelber, New York, 2002.17. M. Takesakai, Theory of operator algebras I, Springer, New York, 1979.18. A. Weil, L'intègration dans les groupes topologiques et ses applications,

Gauthier-Villars, Paris, 1938.

29

Department of Mathematics, Pondicherry University, Pondicherry

605 014

E-mail address: vmuruganandam@gmail.com