dual banach algebras: an overview

Dual Banachalgebras: anoverview

Volker Runde

Dual Banachalgebras

Uniqueness ofthe predual

Representationtheory

Amenability

Virtualdiagonals

Connes-amenability

. . . for vonNeumannalgebras

. . . and ingeneral

Normal, virtualdiagonals

Injectivity

Dual Banach algebras: an overview

Volker Runde

University of Alberta

Goteborg, August 4, 2013


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Dual Banach algebras: the definition

Definition

A dual Banach algebra is a pair (A,A∗) of Banach spaces suchthat:

1 A = (A∗)∗;

2 A is a Banach algebra, and multiplication in A isseparately σ(A,A∗) continuous.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Dual Banach algebras: some examples

Examples

1 Every W ∗-algebra;

2 (M(G ), C0(G )) for every locally compact group G ;

3 (M(S), C(S)) for every compact, semitopologicalsemigroup S ;

4 (B(G ),C ∗(G )) for every locally compact group G ;

5 (B(E ),E ⊗γ E ∗) for every reflexive Banach space E ;

6 Let A be a Banach algebra and let A∗∗ be equipped witheither Arens product. Then (A∗∗,A∗) is a dual Banachalgebra if and only if A is Arens regular;

7 If (A,A∗) is a dual Banach algebra and B is a σ(A,A∗)closed subalgebra of A, then (B,A∗/

⊥B) is a dualBanach algebra.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Uniqueness of the predual, I

Question

Given a dual Banach algebra (A,A∗), is A∗ unique, i.e., if E1

and E2 are Banach spaces such that (A,E1) and (A,E2) aredual Banach algebras, do σ(A,E1) and σ(A,E2) coincide on A?

Theorem (S. Sakai, 1956)

The predual space of a W ∗-algebra is unique.

Theorem (M. Daws, H. L. Pham, S. White, 2009)

Let A be an Arens regular Banach algebra such that A∗∗ isunital. Then A∗ is the unique predual of A.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Uniqueness of the predual, II

But. . .

Example

Let A = `1 with trivial multiplication, i.e., fg = 0 for allf , g ∈ `1. Then (`1, c0) and (`1, c) are both dual Banachalgebras. If σ(`1, c0) and σ(`1, c) coincided on `1, then theinduced images of c0 and c in `∞ would have to coincide. Thisis impossible because the unit ball of c has extreme pointswhereas the one of c0 has none.

Question

Are there “more natural” examples of dual Banach algebraswith non-unique preduals?


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Uniqueness of the predual, III

Indeed. . .

Theorem (M. Daws, et al., 2012)

There is a family (Et)t∈R of Banach spaces such that:

1 (`1(Z),Et) is a dual Banach algebra for each t ∈ R where`1(Z) is equipped with the convolution product;

2 Et∼= c0 for each t ∈ R;

3 σ(`1(Z),Et) 6= σ(`1(Z),Es) for t 6= s.

Still, . . .

Even though the predual A∗ of a dual Banach algebra (A,A∗)need not be unique, there is in many cases a canonical choicefor A∗, e.g., M(G )∗ = C0(G ). In such cases, we usuallysuppose tacitly that we are dealing with that particular A∗, andsimply call A a dual Banach algebra.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Daws’ representation theorem

Recall. . .

(B(E ),E ⊗γ E ∗) is a dual Banach algebra for reflexive E , as iseach of its weak∗ closed subalgebras.

Theorem (M. Daws, 2007)

Let (A,A∗) be a dual Banach algebra. Then there are areflexive Banach space E and an isometric, σ(A,A∗)-weak∗

continuous algebra homomorphism π : A→ B(E ).

In short. . .

Every dual Banach algebra “is” a weak∗ closed subalgebra ofB(E ) for some reflexive E .


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

A bicommutant theorem

Question

Does von Neumann’s bicommutant theorem extend to generaldual Banach algebras?

Example

LetA :=

{[a b0 c

]: a, b, c ∈ C

}.

Then A ⊂ B(C2) is a dual Banach algebra, but A′′ = B(C2).


Let A be a unital dual Banach algebra. Then there are areflexive Banach space E and a unital, isometric, weak∗-weak∗

continuous algebra homomorphism π : A→ B(E ) such thatπ(A) = π(A)′′.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Banach A-bimodules and derivations

Definition

Let A be a Banach algebra, and let E be a Banach A-bimodule.A bounded linear map D : A→ E is called a derivation if

D(ab) := a · Db + (Da) · b (a, b ∈ A).

If there is x ∈ E such that

Da = a · x − x · a (a ∈ A),

we call D an inner derivation.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Amenable Banach algebras

Remark

If E is a Banach A-bimodule, then so is E ∗:

〈x , a · φ〉 := 〈x · a, φ〉 (a ∈ A, φ ∈ E ∗, x ∈ E )

and

〈x , φ · a〉 := 〈a · x , φ〉 (a ∈ A, φ ∈ E ∗, x ∈ E ).

We call E ∗ a dual Banach A-bimodule.

Definition (B. E. Johnson, 1972)

A is called amenable if, for every Banach A-bimodule E , everyderivation D : A→ E ∗, is inner.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Amenability for groups and Banach algebras

Theorem (B. E. Johnson, 1972)

The following are equivalent for a locally compact group G :

1 L1(G ) is an amenable Banach algebra;

2 the group G is amenable.

Theorem (H. G. Dales, F. Ghahramani, & A. Ya. Helemskiı,2002)

The following are equivalent:

1 M(G ) is amenable;

2 G is amenable and discrete.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Amenable C ∗-algebras

Theorem (A. Connes, U. Haagerup, et al.)

The following are equivalent for a C ∗-algebra A:

1 A is nuclear;

2 A is amenable.

Theorem (S. Wasserman, 1976)

The following are equivalent for a von Neumann algebra M:

1 M is nuclear;

2 M is subhomogeneous, i.e.,

M ∼= Mn1(M1)⊕ · · · ⊕Mnk (Mk)

with n1, . . . , nk ∈ N and M1, . . . ,Mk abelian.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Virtual digaonals

Definition (B. E. Johnson, 1972)

An element D ∈ (A⊗γ A)∗∗ is called a virtual diagonal for A if

a ·D = D · a (a ∈ A)

anda∆∗∗D = a (a ∈ A),

where ∆ : A⊗γ A→ A denotes multiplication.

Theorem (B. E. Johnson, 1972)

A is amenable if and only if A has a virtual diagonal.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Normality

Definition (R. Kadison, BEJ, & J. Ringrose, 1972)

Let M be a von Neumann algebra, and let E be a dual BanachM-bimodule. Then:

1 E is called normal if the module actions

M× E → E , (a, x) 7→{

a · xx · a

are separately weak∗-weak∗ continuous;

2 if E is normal, we call a derivation D : M→ E normal if itis weak∗-weak∗ continuous.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Connes-amenability for von Neumann algebras

Theorem (R. Kadison, BEJ, & J. Ringrose, 1972)

Suppose that M is a von Neumann algebra containing a weak∗

dense amenable C ∗-subalgebra. Then, for every normal BanachM-bimodule E , every normal derivation D : M→ E is inner.

Definition (A. Connes, 1976; A. Ya. Helemskiı, 1991)

A von Neumann algebra M is Connes-amenable if, for everynormal Banach M-bimodule E , every normal derivationD : M→ E is inner.

Corollary

If A is an amenable C ∗-algebra, then A∗∗ is Connes-amenable.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Injectivity, semidiscreteness, and hyperfiniteness

Definition

A von Neumann algebra M ⊂ B(H) is called

1 injective if there is a norm one projection E : B(H)→M′

(this property is independent of the representation of Mon H);

2 semidiscrete if there is a net (Sλ)λ of unital, weak∗-weak∗

continuous, completely positive finite rank maps such that

Sλaweak∗−→ a (a ∈M);

3 hyperfinite if there is a directed family (Mλ)λ offinite-dimensional ∗-subalgebras of M such that

⋃λMλ is

weak∗ dense in M.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Connes-amenability, and injectivity, etc.

Theorem (A. Connes, et al.)


1 M is Connes-amenable;

2 M is injective;

3 M is semidiscrete;

4 M is hyperfinite.

Corollary

A C ∗-algebra A is amenable if and only if A∗∗ isConnes-amenable.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Amenability and Connes-amenability, I

The notions of normality and Connes-amenability make sensefor every dual Banach algebra. . .

Proposition

Let A be a dual Banach algebra, and let B be a norm closed,amenable subalgebra of A that is weak∗ dense in A. Then A isConnes-amenable.

Question

Suppose that A is a Connes-amenable, dual Banach algebra.Does it have a norm closed, weak∗ dense, amenable subalgebra?


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Amenability and Connes-amenability, II

Corollary

If A is amenable and Arens regular. Then A∗∗ isConnes-amenable.

Question

Suppose that A is Arens regular such that A∗∗ isConnes-amenable. Is then A amenable?

Theorem (VR, 2001)

Suppose that A is Arens regular and an ideal in A∗∗. Then thefollowing are equivalent:

1 A is amenable;

2 A∗∗ is Connes-amenable.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Amenability and Connes-amenability, III

Corollary

Let E be reflexive and have the approximation property. Thenthe following are equivalent:

1 K(E ) is amenable;

2 B(E ) is Connes-amenable.

Example (N. Grønbæk, BEJ, & G. A. Willis, 1994)

Let p, q ∈ (1,∞) \ {2} such that p 6= q. Then K(`p ⊕ `q) isnot amenable. Hence, B(`p ⊕ `q) is not Connes-amenable.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Normal, virtual diagonals, I

Notation

For a dual Banach algebra A, let B2σ(A,C) denote theseparately weak∗ continuous bilinear functionals on A.

Observations

1 B2σ(A,C) is a closed submodule of (A⊗γ A)∗.

2 ∆∗A∗ ⊂ B2σ(A,C), so that ∆∗∗ : (A⊗γ A)∗∗ → A∗∗ dropsto a bimodule homomorphism ∆σ : B2σ(A,C)∗ → A.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Normal, virtual diagonals, II

Definition (E. G. Effros, 1988 (for von Neumann algebras))

Let A be a dual Banach algebra. Then D ∈ B2σ(A,C)∗ is calleda normal, virtual diagonal for A if

a ·D = D · a (a ∈ A)

anda∆σD = a (a ∈ A).

Proposition

Suppose that A has a normal, virtual diagonal. Then A isConnes-amenable.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Normal, virtual diagonals and Connes-amenability

Question

Is the converse true?

Theorem (E. G. Effros, 1988)

A von Neumann algebra M is Connes-amenable if and only ifM has a normal virtual diagonal.

Theorem (VR, 2003)

The following are equivalent for a locally compact group G :

1 G is amenable;

2 M(G ) is Connes-amenable;

3 M(G ) has a normal virtual diagonal.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Weakly almost periodic functions

Definition

A bounded continuous function f : G → C is called weaklyalmost periodic if {Lx f : x ∈ G} is relatively weakly compact inCb(G ). We set

WAP(G ) := {f ∈ Cb(G ) : f is weakly almost periodic}.

Remark

WAP(G ) is a commutative C ∗-algebra. Its character spaceGWAP is a compact, semitopological semigroup containing Gas a dense subsemigroup. This turns WAP(G )∗ ∼= M(GWAP)into a dual Banach algebra.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Connes-amenability without a normal, virtualdiagonal

Proposition


1 G is amenable;

2 WAP(G )∗ is Connes-amenable.

Theorem (VR, 2006 & 2013)

Suppose that G has small invariant neighborhoods, e.g, iscompact, discrete, or abelian. Then the following areequivalent:

1 WAP(G )∗ has a normal virtual diagonal;

2 G is compact.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Minimally weakly almost periodic groups, I

Definition

A bounded continuous function f : G → C is called almostperiodic if {Lx f : x ∈ G} is relatively compact in Cb(G ). Weset

AP(G ) := {f ∈ Cb(G ) : f is almost periodic}.

We call G minimally weakly almost periodic (m.w.a.p.) if

WAP(G ) = AP(G ) + C0(G ).


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Minimally weakly almost periodic groups, II

Proposition

Suppose that G is amenable and m.w.a.p. Then WAP(G ) hasa normal virtual diagonal.

Examples

1 All compact groups are m.w.a.p.

2 SL(2,R) is m.w.a.p., but not amenable.

3 The motion group RN o SO(N) is m.w.a.p. for N ≥ 2 andamenable.

Question

Does WAP(G )∗ have a normal virtual diagonal if and only ifG is amenable and m.w.a.p.?


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Quasi-expectations

Theorem (J. Tomiyama, 1970)

Let A be a C ∗-algebra, let B be a C ∗-subalgebra of A, and letE : A→ B be a norm one projection, an expectation. Then

E(abc) = a(Eb)c (a, c ∈ B, b ∈ A).

Definition

Let A be a Banach algebra, and let B be a closed subalgebra.A bounded projection Q : A→ B is called a quasi-expectationif

Q(abc) = a(Qb)c (a, c ∈ B, b ∈ A).


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Quasi-expectations and injectivity

Theorem (J. W. Bunce & W. L. Paschke, 1978)

The following are equivalent for a von Neumann algebraM ⊂ B(H):

1 M is injective;

2 there is a quasi-expectation Q : B(H)→M′.

“Definition”

We call a dual Banach algebra A “injective” if there are areflexive Banach space E , an isometric, weak∗-weak∗

continuous algebra homomorphism π : A→ B(E ), and aquasi-expectation Q : B(E )→ π(A)′.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Connes-amenability and injectivity, I

Easy. . .

Connes-amenability implies “injectivity”, but. . .

Example

For p, q ∈ (1,∞) \ {2} with p 6= q, B(`p ⊕ `q) is notConnes-amenable, but trivially “injective”.

Definition (M. Daws, 2007)

A dual Banach algebra A is called injective if, for each reflexiveBanach space E and for each weak∗-weak∗ continuous algebrahomomorphism π : A→ B(E ), there is a quasi-expectationQ : B(E )→ π(A)′.


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

Connes-amenability and injectivity, II


The following are equivalent for a dual Banach algebra A:

1 A is injective;

2 A is Connes-amenable.

Question

Let A be a Connes-amenable, dual Banach algebra, let E be areflexive Banach space, and let π : A→ B(E ) be an isometric,weak∗-weak∗ continuous algebra homomorphism. Is there aquasi-expectation Q : B(E )→ π(A)?


Volker Runde

Dual Banachalgebras



Amenability

Virtualdiagonals

Connes-amenability


. . . and ingeneral


Injectivity

And now something completely (not quite)different. . .

THEMATIC PROGRAMon

ABSTRACT HARMONIC ANALYSIS,BANACH, AND OPERATOR ALGEBRAS

at the

Fields Institute(http://www.fields.utoronto.ca)

January–June, 2014

dual banach algebras: an overview

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