In memoriam Charles John Read – mathematician, gentleman…pmt6jrp/ReadSlides/loy_slides.pdf ·...

In memoriam Charles John Read – mathematician, gentleman, and friend

Rick Loy (Australian National University) 1 / 44

Approximate amenability of tensor products ofBanach algebras

Rick Loy

Mathematical Sciences InstituteAustralian National University

September 2016, University of Leeds

This is joint work with Fereidoun Ghahramani


Outline

1 Reminders from Fereidoun’s talk

2 Tensor products

3 Semi-inner derivations

4 Applications to tensor products


Basic reminders from Fereidoun’s talkThroughout A will be a Banach algebra, X a Banach A-bimodule.

DefinitionA derivation D : A→ X is a linear map such that

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

All derivations are assumed to be continuous.

A derivation D : A→ X is inner if, for some ξ ∈ X , it is of the form

adξ : a 7→ a · ξ − ξ · a (a ∈ A) ,

andit is approximately inner if, for some net (ξi) ⊂ X ,

D(a) = lim adξi (a) (a ∈ A) .


More reminders

DefinitionA is amenable if for every A-bimodule X any derivationD : A→ X ∗ is inner.A is approximate amenable if for every A-bimodule X anyderivation D : A→ X ∗ is approximately inner.

Fact [G, L, Read & Zhang]Approximate amenability⇐⇒ weak∗ approximate amenability⇐⇒derivations into any Banach bimodule are approximately inner.

The last property here is approximate contractibility .


Last reminders

DefinitionA is boundedly approximate amenable if for every A-bimodule X anyderivation D : A→ X ∗ is approximately inner with a bounded net ofapproximating inner derivations.

NB. This is not the same if one allows any A-bimodule.

NB. It is the net of approximating inner derivations that is required tobe bounded, not the net of implementing elements.

Boundedness of the implementing net is a much stronger condition:

Theorem (Gourdeau)The Banach algebra A is amenable if and only if any derivationD : A→ X ∗ into a dual bimodule is approximable by a net (adξi ) withthe net (ξi) bounded.


Some examples

Any finite dimensional approximately amenable algebra isamenable.

For any locally compact space X , C(X ) is amenable. (Note thatan amenable uniform algebra A must be C(ΦA), whether the sameholds for approximate amenability is unknown.)

For any locally compact group G, L1(G) is approximatelyamenable if and only if it boundedly approximately amenable ifand only if it is amenable, if and only if G is amenable as a group.

c0((`1n)#) is approximately amenable but not amenable.

`1(N), c0(`1n) are neither.

Note that c0((`1n)#) has a bounded approximate identity, c0(`1n)does not.


c0-sums

Much more work is required to split boundedly approximatelyamenable and approximately amenable.

Theorem (G & Read)There exists a sequence (An) of boundedly approximately amenablealgebras such that c0(An) is approximately amenable but notboundedly approximately amenable.

The approximate amenability of c0(An) here is a non-trivial part of theresult – cf. c0(`1n) above.

In fact things can go ‘awry’ with finite sums.


More on sums

Theorem (G & Read)There exists a boundedly approximately amenable algebra A such thatA⊕ Aop is not approximately amenable.

(Here A = c0(An) with (An) amenable, whence A⊕ Aop has this sameform. Also, A⊕ A is boundedly approximately amenable.)

There is one way to avoid this ‘pathology’ for finite direct sums:

Theorem (G, L & Zhang)If A and B are approximately amenable, and one of them has abounded approximate identity, then A⊕ B is approximately amenable.

So again, the presence of a bounded approximate identity facilitates‘good behaviour’.


A little building

The above examples/results with sums are more than just usefulindicators as we now turn to the topic at hand, namely tensor products.

Theorem (Johnson)The tensor product of amenable algebras is again amenable.

Barry’s argument can be used more generally:

Theorem (Choi, G & L)Suppose that A is approximately amenable with a boundedapproximate identity, that B is amenable, and let D be a derivation fromA ⊗̂B to a dual bimodule. Then D is approximately inner on A⊗ B.

If A is boundedly approximately amenable then A ⊗̂B is boundedlyapproximately amenable.


A little building

So, for instance, with the strong additional hypotheses:

A has a bounded approximate identity,B is finite dimensional amenable,

A approximately amenable implies A ⊗̂B is approximately amenable.

Overkill?

Well, as you probably suspect by now . . .

The tensor product of boundedly approximately amenablealgebras need not be approximately amenable.


Example

Let A be the Banach algebra of G & R such that A is boundedlyapproximately amenable yet A⊕ Aop is not approximately amenable.

Write B = Aop, adjoin identities 1A to A and 1B to B, and setA = A# ⊗̂B#.

Then A decomposes into closed subspaces:

A = C(1A ⊗ 1B) + (C1A ⊗ B) + (A⊗ C1B) + (A ⊗̂B) .

Here A ⊗̂B is a closed two-sided ideal; consider A/(A ⊗̂B).


Example continued

If A is approximately amenable, then so is the quotient algebra

C(1A ⊗ 1B) + (C1A ⊗ B)⊕ (A⊗ C1B) '(

(C1A ⊗ B)⊕ (A⊗ C1B))#

,

whence so is (C1A ⊗ B)⊕ (A⊗ C1B) ' A⊕ B .

But by the specific choice of A and B, A⊕ B is not approximatelyamenable.

So A cannot be approximately amenable.


UnitizationsA similar argument (with same A and B) shows that for the boundedlyapproximately amenable algebra A = A# ⊕ B#, A⊗̂A is notapproximately amenable.

Unitizations are used here to obtain counterexamples. They are alsouseful in giving positive results.

TheoremSuppose that A# ⊗̂B# is approximately amenable. Then A, B andA⊕ B are approximately amenable.

Proof.Both A# and B# are homomorphic images. For A⊕ B use anotherdecomposition argument.

Something missing here?Rick Loy (Australian National University) 14 / 44

Bounded approximate identities

TheoremSuppose that A# ⊗̂B# is (boundedly) approximately amenable, andthat A and B have bounded approximate identities. Then A ⊗̂B is(boundedly) approximately amenable.

Proof.Thanks to the bounded approximate identities, it suffices to considerneo-unital (A ⊗̂B)-bimodules.

Thanks to the bounded approximate identities again, any derivationD : A ⊗̂B → X ∗ lifts to the double centralizer algebra, then restricts toA# ⊗̂B#, where it is suitably approximately inner.


More bounded approximate identities

TheoremLet A and B be Banach function algebras on their respective carrierspaces ΦA and ΦB, and suppose that A and B have boundedapproximate identities consisting of elements of finite support. ThenA ⊗̂B is approximately amenable.

Proof 1.The carrier space of A ⊗̂B is ΦA × ΦB, which is discrete.

The bounded approximate identities having finite support means that Aand B have the (bounded) approximation property, so that the naturalmap A ⊗̂B → A ⊗∧B is injective.

It follows that A ⊗̂B is semisimple, and so is a Banach function algebraon ΦA × ΦB.


More bounded approximate identities

Proof 2.Further, A ⊗̂B has a bounded approximate identity consisting ofelements of finite support, built from those of A and B.

But it is well known that a Banach function algebra having a boundedapproximate identity consisting of elements of finite support isapproximately amenable.

(There are several published proofs; all essentially the same except fortheir generality.)

So A, B and A ⊗̂B are approximately amenable.


Semi-inner derivations

A new name for something well-known!

DefinitionLet A be a algebra, X an A-bimodule. A map D : A→ X is semi-inner ifthere are m,n ∈ X such that

D(a) = a ·m − n · a (a ∈ A) .

Such maps, with B a superalgebra of A, are commonly known as‘generalized inner derivations’ .

When D is also a derivation, then m and n are highly constrained:

a · (m − n) · b = 0 (a,b ∈ A) .

In the Banach case, with D : A→ X ∗ with X neo-unital, thennecessarily m = n and D is inner.


Semi-inner derivations

To us ‘approximately generalized’ is an oxymoron, so we use‘semi-inner’, and only use it for derivations:

DefinitionFor A a Banach algebra, X a Banach A-bimodule, a derivationD : A→ X is approximately semi-inner if there are nets (mi), (ni) in Xwith

D(a) = limi

(a ·mi − ni · a) (a ∈ A) .

In this case, for X neo-unital, and D : A→ X ∗, then

limi

(a · (mi − ni) · b

)= 0 (a,b ∈ A) ,

whence mi − ni → 0 weak∗, so that D is in fact weak∗ approximatelyinner, and hence approximately inner by the Fact.


Example

Take A = `2 under pointwise operations, D : `2 → X a derivation intoan A-bimodule, (En) the standard (unbounded) approximate identity of`2: En = (1,1, . . . ,1︸︷︷︸

n times

,0,0, . . .).

The map Dn : En`2 → X is a derivation from a finite-dimensional

semisimple algebra and hence is inner, say implemented by ξn ∈ X .

Thus for a ∈ `2,

D(a) = limn

D(Ena) = limn

(Ena · ξn − ξn · Ena)

= limn

(a · (En · ξn)− (ξn · En) · a

),

So D is approximately semi-inner.

Note that `2 is not approximately amenable.


A simple construction

Let A and B be Banach algebras, D : A→ X a derivation into anA-bimodule X .

Make X ⊗̂B into an A ⊗̂B-bimodule as follows: for a ∈ A, b1,b2 ∈ Band x ∈ X , set

(a⊗ b1) · (x ⊗ b2) = a · x ⊗ b1b2 , (x ⊗ b2) · (a⊗ b1) = x · a⊗ b2b1 .

The map ∆ : A ⊗̂B → X ⊗̂B defined by

∆(a⊗ b) = D(a)⊗ b (a ∈ A,b ∈ B) .

is a derivation.

Fix b0 ∈ B, b∗0 ∈ B∗ with 〈b∗0,b0〉 = 1 and define the operator

T : X ⊗̂B → X : x ⊗ b 7→ 〈b∗0,b〉x .


Lemma (A)

Lemma (A)

Suppose that A ⊗̂B is approximately amenable. Then any derivationfrom A or B into any Banach bimodule is approximately semi-inner.

Proof. (For the algebra A)

Given a derivation D : A→ X , take ∆ : A ⊗̂B → X ⊗̂B as above. Byapproximate amenability ∆ is approximately inner.

(This uses the Fact that approximate amenability is the same asapproximate contractibility.)


Lemma (A)

Proof cont.

So there is a net(∑∞

k=1 xk ,i ⊗ bk ,i

)iin X ⊗̂B with

∆(a⊗ b) = limi

(∑k

(a · xk ,i)⊗ bbk ,i −∑

k

(xk ,i · a)⊗ bk ,ib

).

Applying T to both sides,

D(a) = limi

(a ·m′i − n′i · a) ,

where m′i =∑

k 〈b∗0,b0bk ,i〉xk ,i , n′i =∑

k 〈b∗0,bk ,ib0〉xk ,i .


Lemma (B)

Lemma (B)

Suppose that A ⊗̂B is boundedly approximately amenable. Then anyderivation from A or B into any dual bimodule is boundedlyapproximately semi-inner.

NB Conclusion is both weaker and stronger than Lemma (A).

Proof 1.We cannot argue as before! And X ∗ ⊗̂B is unlikely to be dual.

We start with a more sophisticated version of the map T .


Lemma (B)Proof 2.Fix b0 ∈ B, b∗0 ∈ B∗ with b∗0(b0) = 1, and let S : X → (X ∗ ⊗̂B)∗ bespecified by

〈S(x), x∗ ⊗ b〉 = 〈x∗, x〉〈b∗0,b〉 , (x ∈ X ,b ∈ B) ,

and set T = S∗ : (X ∗ ⊗̂B)∗∗ → X ∗.

For m =∑

k x∗k ⊗ bk ∈ X ∗ ⊗̂B, a straightforward calculation yields

T ((a⊗ b0) ·m) =∑

k

〈b∗0,b0bk 〉a · x∗k = a ·∑

k

〈b∗0,b0bk 〉x∗k︸︷︷︸=:x∗(m)

,

with the estimate‖x∗(m)‖ ≤ ‖b0‖ ‖b∗0‖ ‖m‖ .


Lemma (B)

Proof 4.For a general m ∈ (X ∗ ⊗̂B)∗∗, a weak∗-density and compactness

argument, using the weak∗ to weak∗-continuity of T , now shows that

there is ξ∗ ∈ X ∗, bounded by a multiple of ‖m‖, which satisfies

T ((a⊗ b0) ·m) = a · ξ∗ (a ∈ A) .

Similarly, there is η∗ ∈ X ∗, bounded by a multiple of ‖m‖, with

T (m · (a⊗ b0)) = η∗ · a (a ∈ A) .


Lemma (B)

Proof 5.Now back to derivations.

Given a derivation D : A→ X , take ∆ : A ⊗̂B → X ⊗̂B as above,viewed as mapping into (X ∗ ⊗̂B)∗∗.

Then there is a net (mi) in (X ∗ ⊗̂B)∗∗ and a constant K > 0 such thatfor a ∈ A,b ∈ B,

D(a)⊗ b = ∆(a⊗ b) = limi

((a⊗ b) ·mi −mi · (a⊗ b)

),

‖(a⊗ b) ·mi −mi · (a⊗ b)‖ ≤ K‖a‖ ‖b‖ .


Lemma (B)

Proof 6.

Setting b = b0, and applying T gives nets (m′i ) and (n′i ) in X ∗ with

D(a) = limi

(a ·m′i − n′i · a) (a ∈ A) ,

‖a ·m′i − n′i · a‖ ≤ K‖T‖ ‖b0‖‖a‖ (a ∈ A) .

Since D is a derivation one also gets the bounds

‖a · (m′i − n′i ) · b‖ ≤ 3K‖T‖‖b0‖‖a‖‖b‖ , (a,b ∈ A) ,

which we will need later.


When m′i = n′i

TheoremSuppose that A ⊗̂B is (boundedly) approximately amenable. If B hasan element b0 with b0 6∈ {b0b − bb0 : b ∈ B} , then A is (boundedly)approximately amenable.

Proof.Choose the functional b∗0 in the proof of the Lemmas to vanish on{b0b − bb0 : b ∈ B}. Then the resulting nets (m′i ) and (n′i ) are thesame.

The commutator condition here is due to Barry Johnson.

Note there is no conclusion about B.


When m′i − n′i → 0

TheoremSuppose that A ⊗̂B is boundedly approximately amenable. Supposethat one of A or B has an identity. Then A and B are boundedlyapproximately amenable.

Proof 1.Suppose that B has an identity e. Previous result gives A isapproximately amenable. But what about B?


When m′i − n′i → 0

Proof 2.Let X be a Banach B-bimodule. By the usual reduction, we maysuppose that e · x = x = x · e for x ∈ X .

Let D : B → X ∗ be a derivation, and consider the nets given by Lemma(B). Since

limi

(b · (m′i − n′i ) · c

)= 0 (b, c ∈ B) ,

putting b = c = e gives m′i − n′i → 0, so that D is boundedlyapproximately inner.


Use of bounded approximate identities

TheoremSuppose that A ⊗̂B is boundedly approximately amenable and that Ahas a bounded approximate identity. Then A is approximatelyamenable.

Proof 1.Let D : A→ X ∗ be a derivation into the dual of a neo-unital bimoduleX . From Lemma (B), we have nets (m′i ), (n′i ) in X ∗, and K ≥ 0 suchthat for a,a1,a2 ∈ A

D(a) = limi

(a ·m′i − n′i · a) , ‖a ·m′i − n′i · a‖ ≤ K‖a‖ , (1)

limi

(a1 · (m′i − n′i ) · a2) = 0 , ‖a1 · (m′1 − n′i ) · a2‖ ≤ 3K‖a1‖ · ‖a2‖ . (2)



Proof 2.

For a given x ∈ X , (2) gives

〈m′i − n′i ,a2xa1〉 → 0, |〈m′i − n′i ,a2xa1〉| ≤ 3K‖a1‖ · ‖a2‖ · ‖x‖ .

Since X is neo-unital, it follows that

m′i − n′i → 0 weak∗ ,

and letting a1,a2 range over an approximate identity with bound M ,

‖m′i − n′i‖ ≤ 3KM2 .



Proof 3.Together with (1), these give for a ∈ A,

D(a) = weak∗ − limi

(a ·m′i −m′i · a) , ‖a ·m′i −m′i · a‖ ≤ 4M2‖a‖ ,

and we have derivations from A into duals of neo-unital bimodules areboundedly weak∗- approximately inner.

It remains to remove the neo-unital assumption.

This is a standard decomposition argument for approximation in norm.

The same type of argument also works here, but only because of theboundedness.


Lemma (C)

Lemma (C)Let A have a bounded approximate identity. Suppose that anyderivation from A into the dual of a neo-unital bimodule is boundedlyweak∗-approximately inner.

Then A is (boundedly) weak∗-approximately amenable, and soapproximately amenable.

Proof 1.Let (eα) be a bounded approximate identity for A. Let E be aweak∗-limit point of the left multiplication operators on X ∗ by theelements of (eα), F similarly for right multiplication.


Lemma (C)

Proof 1.Then E and F are commuting projections on X ∗, and give adecomposition

X ∗ = EFX ∗ ⊕ E(I − F )X ∗ ⊕ (I − E)X ∗ .

For a derivation D : A→ X ∗, set

D1 = EFD,D2 = E(I − F )D,D3 = (I − E)D .

Then D1,D2,D3 are derivations into the corresponding summands ofX ∗.


Lemma (C)

Proof 2.The actions of A on the right of E(I − F )X ∗, and on the left of

(I − E)X ∗, are trivial, and A has a bounded approximate identity, so

that D2 and D3 are boundedly approximately inner.

For D1, note that EFX ∗ is isomorphic to (Xess)∗, where Xess = A · X · Ais a neo-unital A-bimodule. So by hypothesis, D1 : A→ EFX ∗ is

boundedly weak∗-approximately inner.


Lemma (C)

Proof 3.A consideration of the weak∗ topologies on the bounded sets of EFX ∗

and X ∗ shows that D1 : A→ X ∗ is boundedly weak∗-approximatelyinner.

Adding D1,D2 and D3, D is boundedly weak∗-approximately inner, andhence approximately inner by Fact.


Central bounded approximate identities

TheoremSuppose that A ⊗̂B is approximately amenable and that one of A or Bhas a central bounded approximate identity. Then A and B areapproximately amenable.

Proof 1.Suppose that (eα) is a central bounded approximate identity in B. LetD : B → X ∗ be a derivation into the dual of a neo-unital bimodule X .

From Lemma (A), we have a nets (m′i ) and (n′i ) in X ∗ such that

D(b) = limi(b ·m′i − n′i · b) (b ∈ B) ,

limi(b1 · (m′i − n′i ) · b2) = 0 (b1,b2 ∈ B) .



Proof 2.Now follow Lemma (C) to get D1,D2 and D3. Then for b ∈ B,

D1(b) = (w∗ − limα

)(w∗ − limβ

)eαD(b)eβ

= (w∗ − limα

)(w∗ − limβ

) limi

[eα(b ·m′i − n′i · b)eβ] .

Using centrality of (eα),

D1(b) = (w∗ − limα

)(w∗ − limβ

) limi

[b ·︷︸︸︷(eα ·m′i · eβ)−

︷︸︸︷(eα · n′i · eβ) ·b] .



Proof 3.Thus the standard method of considering finite subsets of B and X ,gives a net (x∗γ ) ⊂ X ∗ such that

D1(b) = weak∗ − limγ

(b · x∗γ − x∗γ · b) , (b ∈ B) .

Since D2 and D3 are approximately inner we finally deduce that D isweak∗-approximately inner. Thus B is approximately amenable.

Since B has a non-zero central element, A is approximately amenableby the Johnson criterion.


Extension of Barry’s result

Theorem

Suppose that A ⊗̂B is amenable. Then A and B are amenable.

Proof 1.Amenability of A ⊗̂B implies it has a bounded approximate identity,whence so do A and B.

Let D : A→ X ∗ be a derivation into the dual of a neo-unital bimoduleX .

Take ∆ as in Lemma (A) and use the necessary part of Gourdeau’stheorem to obtain a bounded net (mi). This then gives a bounded nets(m′i ) and (n′i ) with

D(a) = limi

(a ·m′i − n′i · a) (a ∈ A) .


Extension of Barry’s result

Proof 2.The argument of a previous theorem gives D boundedlyweak∗-approximately inner, with implementing net bounded.

Now use the argument of Lemma (C) to see that derivations from Ainto a dual module are weak∗-approximately inner, with a bounded netof implementing elements. The argument behind the Fact now showsthat any derivation into any A-bimodule is approximately inner with abounded net of implementing elements, that is, A is amenable by thesufficient part of Gourdeau’s theorem.


Three questions

These obvious questions remain :1 Does A# ⊗̂B# (boundedly) approximately amenable imply A ⊗̂B

(boundedly) approximately amenable? (Yes, with a boundedapproximate identity in each factor)

2 Does A ⊗̂B (boundedly) approximately amenable imply A and Bare (boundedly) approximately amenable? (Yes, with a centralbounded approximate identity in either factor)

3 Does A ⊗̂B (boundedly) approximately amenable imply A# ⊗̂B#

(boundedly) approximately amenable?

THANK YOU FOR YOUR ATTENTION


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