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arXiv:0908.3566v1 [math.FA] 25 Aug 2009

APPROXIMATE CONNES-AMENABILITY OF

DUAL BANACH ALGEBRAS

G. H. Esslamzadeh and B. Shojaee

Abstract. We introduce the notions of approximate Connes-amenability and

approximate strong Connes-amenability. Then we characterize these two types

of dual Banach algebras in terms of approximate normal virtual diagonal and

approximate σWC− virtual diagonals. Some concrete cases are also discussed.

Mathematics Subject Classification (2000). Primary 46H25, 46H20; Secondary

46H35.

Keywords. Approximately inner derivation, Approximately weakly amenable,

Approximately amenable, Approximate trace extension property.

1. INTRODUCTION

In [9], B.E. Johnson, R. V. kadison, and J. Ringrose introduced a notion of

amenability for von neuman algebras which modified Johnson’s original defini-

tion for Banach algebras[8] in the sense that it takes the dual space structure

of a von Neumann algebra into account. This notion of amenability was later

called Connes-amenability by A. Ya. Helemskii [7]. In [10], the author extended

the notion of Connes-amenability to the larger class of dual Banach algebras. The

concept of approximate amenability for a Banach algebras was introduced by F.

Ghahramani and R. J. Loy in [5]. Our motivation for introducing approximate

Connes-amenability is finding a version of these approximate forms of amenablity

which sounds suitable for dual Banach algebras.

Before proceeding further we recall some terminology.

The main part of this work was undertaken while the second author was studying for PhD at

the Islamic Azad University. The second author wishes to thank Islamic Azad University for the

financial support.

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2 G. H. Esslamzadeh and B. Shojaee

Throughout A is a Banach algebra and X is a Banach A-bimodule. A derivation

is a bounded linear map D : A −→ X such that

D(ab) = a.Db + Da.b(a,b ∈ A).

For x ∈ X, define

δx(a) = a.x − x.a(a ∈ A).

Then δxis a derivation; maps of this form are called inner derivation. A derivation

D : A −→ X is approximately inner if there exists a net (aα) ⊆ X such that for

every a ∈ A, D(a) = limα(a.xα−xα.a), the limit being in norm. We say that A is

approximately amenable if for any A-bimodule X, every derivation D : A −→ X∗

is approximately inner. A is called a dual Banach algebra if there is a closed

submodule A∗of A∗such that A = (A∗)∗. For example if G is a locally compact

group, then M(G) is a dual Banach algebra(with A∗= c0(G)), or if A is an Arens

regular Banach algebra then A∗∗is a dual Banach algebra.

Let A be a dual Banach algebra. A dual Banach A-bimodule X is called normal

if, for each x ∈ X, the maps

A −→ X,a ?−→

?

a.x

x.a

are ω∗-continuous. A dual Banach algebra A is Connes-amenable if, for every

normal, dual Banach A-module X , every ω∗-continuous derivation D : A −→ X

is inner.

Let A be a dual Banach algebra, and let X be a Banach A-bimodule. Then we

call an element φ ∈ X∗a ω∗-element if the maps

A −→ X∗,a ?−→

?

a.φ

φ.a

are ω∗− ω∗continuous.

2. APPROXIMATE CONNES AMENABILITY

Definition 2.1. A dual Banach algebra A is approximately Connes-amenable if for

every normal, dual Banach A-bimodule X, every ω∗-continuous derivation D ∈

Z1(A,X) is approximately inner.

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APPROXIMATE CONNES-AMENABILITY OF DUAL BANACH ALGEBRAS 3

Lemma 2.2. Suppose that A is approximately Connes-amenable. Then A has left

and right approximate identities. In particular A2is dense in A.

Proof. Let X be the Banach A-bimodule whose underlying linear space is A

equipped with the following module operation:

a.x = axandx.a = 0(a ∈ A,x ∈ X).

Obviously X, is a normal dual Banach A-bimodule and the identity map on A

is a ω∗− continuous derivation. Since A is approximately Connes-amenable, then

there exists a net (aα) ⊆ X such that

a = lim

αaaα

(a ∈ A).

This means that A has a right approximate identity. Similarly, one see that A has

a left approximate identity.

?

Let (A,A∗) be a dual Banach algebra, and let A#be the Banach algebra A ⊕ C.

Then A#is a dual Banach algebra with predual A∗⊕ C and norm

? (µ,α) ?= max(?µ?,|α|)(µ ∈ A∗,α ∈ C).

Proposition 2.3. Let A be a dual Banach algebra. A is approximately Connes-

amenable if and only if A#is approximately Connes-amenable.

Proof. Let D : A#−→ X∗be a ω∗-continuous derivation for normal dual Banach

A#-bimodule X∗. By [5, Lemma 2.3], D = D1+ adη where D1: A#−→ e.X∗.e

is a ω∗-continuous derivation and η ∈ X∗. Since e.X∗.e is a normal dual Ba-

nach A-bimodule, then D1(e) = 0 and D1|Ais approximately inner; whence D is

approximately inner. Thus A#is approximately Connes-amenable.

Now suppose D : A −→ X∗is a ω∗-continuous derivation for normal dual Banach

A-bimodule X∗. Set

?D : A#−→ X∗,

?D(a + λe) = Da

(e ∈ A#,x ∈ X), then X∗turns into a normal dual

(a ∈ A,λ ∈ C).

If we define e.x = x.e = x

Banach A#-bimodule and?D is a ω∗-continuous derivation. So?D is approximately

inner, and hence so is D. It follows that A is approximately Connes-amenable.

?

Proposition 2.4. Let A be an Arens regular dual Banach algebra. If A∗∗is approx-

imately Connes-amenable, then A is approximately Connes-amenable.

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4 G. H. Esslamzadeh and B. Shojaee

Proof. Suppose X is a normal dual Banach A-bimodule, and π : A∗∗−→ A is the

restriction map to A∗. Then π is a ω∗−ω∗continuous homomorphism. Therefore

X is a normal dual Banach A∗∗-bimodule with the following actions

a∗∗.x = π(a∗∗)x,x.a∗∗= xπ(a∗∗)(x ∈ X,a∗∗∈ A∗∗).

Let D : A −→ X be a ω∗− ω∗continuous derivation. It is easy to see that

Doπ : A∗∗−→ X is a ω∗− ω∗continuous derivation. Since A∗∗is approximately

Connes-amenable, than there exists a net (xα) ⊆ X such that

Doπ(a∗∗) = lim

αa∗∗.xα− xα.a∗∗

(a∗∗∈ A∗∗).

So

D(a) = lim

αa.xα− xα.a(a ∈ A).

?

For a locally compact group G we have:

Theorem 2.5. If L1(G) is approximately amenable then M(G) is approximately

Connes-amenable.

Proof. By [5, Theorem 3.2], G is amenable, and by [11, Theorem 4.4.13], M(G) is

approximately Connnes-amenable.

?

Theorem 2.6. Let A be an Arens regular Banach algebra which is an ideal in

A∗∗. If A∗∗is approximately Connes-amenable and has an identity then A is

approximately amenable.

Proof. By [5, Proposition 2.5] in order to show that A is approximately amenable

it is sufficient to show that every D ∈ Z1(A,X∗) is approximately inner for each

neo-unital Banach A-module.

Let X be a neo-unital Banach A-bimodule, and let D ∈ Z1(A,X∗). By [11, Theo-

rem 4.4.8] X∗is a normal dual Banach A∗∗-bimodule and D has a unique extension

?D ∈ Z1(A∗∗,X∗). From the approximate Connes-amenability of A∗∗we conclude

that?D, and hence D is inner. It follows that A is approximately amenable.

In the following proposition we obtain a criterion for approximate amenability

?

which will be used in the sequel.

Proposition 2.7. The following conditions are equivalent;

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APPROXIMATE CONNES-AMENABILITY OF DUAL BANACH ALGEBRAS 5

(i) A is approximately amenable;

(ii) For any A-bimodule X, every bounded derivation D : A −→ X∗∗is approxi-

mately inner.

Proof (i) =⇒ (ii) This is immediate.

(ii) =⇒ (i). By [6, Theorem 2.1], it suffices to show that if D ∈ Z1(A,X), then it

is approximately inner.

We have ι ◦ D ∈ Z1(A,X∗∗), where ι : X −→ X∗∗is the canonical embedding.

Thus, there exists a net (x∗∗

α) ⊆ X∗∗such that

ι ◦ D(a) = lim

αa.x∗∗

α− x∗∗

α.a(a ∈ A).

Now take ǫ > 0, and finite sets F ⊆ A, φ ⊆ X∗.

Then there is α such that

|?x∗,ι ◦ D(a) − (a.x∗∗

α− x∗∗

α.a)?| < ǫ

For all x∗∈ φ and a ∈ F.

By Goldstine’s Theorem, there is a net (xα) in X such that

|?x∗,ι ◦ D(a) − (a.xα− xα.a)?| < ǫ

for all x∗∈ φ and a ∈ F. Thus there is a net (xα) ⊆ X such that

Da = ω − lim

αa.xα− xα.a(a ∈ A).

Finally, for each finite set F ⊆ A, say F = {a1,...,an},

(a1.xα− xα.a1,...,an.xα− xα.an) −→ (Da1,...,Dan)

weakly in (X)n. By Mazure’s Theorem,

(Da1,...,Dan) ∈ Co?.?{(a1.xα− xα.a1,...,an.xα− xα.an)},

Thus there is a convex linear combination xF,ǫof elements in the set {xα} such

that ,

?Da − (a.x(F,ǫ)− x(F,ǫ).a)? < ǫ(a ∈ F).

The family of such pairs (F,ǫ) is a directed set for the partial order ≤ given by

(F1,ǫ1) ≤ (F2,ǫ2)ifF1⊆ F2

andǫ1≥ ǫ2.

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6 G. H. Esslamzadeh and B. Shojaee

Thus

Da = lim

(F,ǫ)a.x(F,ǫ)− x(F,ǫ).a(a ∈ A).

?

One might ask whether the ideal condition in the Theorem 2.6 is necessary for A.

We answer this question, partially.

Theorem 2.8. Let A be an Arens regular Banach algebra which is a right ideal in

A∗∗. Let for every A∗∗-bimodule X, X∗A = X∗. If A∗∗is approximately Connes-

amenable, then A is approximately amenable.

Proof. Let X be a Banach A-bimodule. By proposition 2.7 it suffices to show that

every D ∈ Z1(A,X∗∗) is approximately inner.

Let D ∈ Z1(A,X∗∗). By [2, p. 27], X∗∗∗∗is a Banach A∗∗-bimodule and by [1,

Proposition 2.7.17(i)],

D∗∗: A∗∗−→ X∗∗∗∗

is a ω∗− ω∗continuous derivation. Since X∗∗∗is an A∗∗-bimodule, then by as-

sumption

X∗∗∗∗A = X∗∗∗∗.(1)

We claim that X∗∗∗∗is a normal dual Banach A∗∗-bimodule. Let (a

′′

α) be a net in

A∗∗such that a

′′

α−→ω∗a

′′. Then

aa

′′

α−→ω∗aa

′′

(a ∈ A).

Since A is a right ideal of A∗∗and ω∗-topology of A∗∗restricted to A coincides

with the weak topology, we have

aa

′′

α−→ωaa

′′

(a ∈ A).

Let x

′′′′∈ X∗∗∗∗. Then by (1) there exists a ∈ A and y

′′′′∈ X∗∗∗∗such that

x

′′′′= y

′′′′a. Thus we have,

x

′′′′a

′′

α= y

′′′′.aa

′′

α−→ωy

′′′′.aa

′′= x

′′′′a

′′.

Therefore

x

′′′′a

′′

α−→ω∗x

′′′′a

′′.

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APPROXIMATE CONNES-AMENABILITY OF DUAL BANACH ALGEBRAS 7

On the other hand by definition,

a

′′

αx

′′′′−→ω∗a

′′x

′′′′

Now since A∗∗is approximately Connes-amenable, then there exists a net (x

′′′′

α) ∈

X

′′′′, such that

D∗∗a

′′= lim

αa

′′.x

′′′′

α− x

′′′′

α.a

′′

(a

′′∈ A∗∗).

Let P : X

′′′′−→ X

′′be the natural projection, then

Da = lim

αa.P(x

′′′′

α) − P(x

′′′′

α).a(a ∈ A).

Thus D is approximately inner. It follow that A is approximately amenable.

?

Proposition 2.9. Suppose that A is a dual Banach algebra with identity. Then A

is approximately Connes-amenable if and only if every ω∗−continuous derivation

into every unital normal dual Banach bimodule X is approximately inner.

Proof. (=⇒): Obvious.

(⇐=) Suppose D ∈ Z1(A,X) is a ω∗− ω∗continuous derivation into the nor-

mal dual Banach bimodule E. By [5, Lemma 2.3], we have D = D1+ adη where

D1: A −→ e.X.e is a derivation and η ∈ X∗. Since D is a ω∗−continuous deriva-

tion and X is a normal dual Banach bimodule then D1 is ω∗−continuous and

e.X.e is normal. So by assumption D1is approximately inner, and therefore A is

approximately Connes-amenable.

?

Let A be a dual Banach algebra with identity, and let L2

ω∗(A,C) be the space of

separately ω∗− ω∗-continuous bilinear functionals on A. Clearly, L2

ω∗(A,C) is a

Banach A-submodule of L2(A,C) and

L2(A,C) ≃ (A? ⊗A)∗.

From [10] we have a natural A-bimodule map

θ : A ⊗ A −→ L2

ω∗(A,C)∗

defined by letting θ(a ⊗ b)F = F(a,b). Since A∗⊗ A∗⊆ L2

ω∗(A,C) and A∗⊗ A∗

separates points of A ⊗ A, then θ is one-to-one. We will identify A ⊗ A with its

image, writing

A ⊗ A ⊆ L2

ω∗(A,C)∗.

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8 G. H. Esslamzadeh and B. Shojaee

The map ∆Ais defined as follow:

∆A: A? ⊗A −→ A,a ⊗ b ?−→ ab(a,b ∈ A)

Since multiplication in a dual Banach algebra is separately ω∗− ω∗-continuous,

we have,

∆∗

AA∗⊂ L2

ω∗(A,C),

So that restriction, of ∆∗∗

Ato L2

ω∗(A,C) turns into a Banach A-bimodule homo-

morphism

∆ω∗ : L2

ω∗(A,C)∗−→ A

Suppose F ∈ L2

ω∗(A,C) and M ∈ L2

?

ω∗(A,C)∗. We have the notation,

?

F(a,b)dM(a,b) = FdM := ?M,F?.

More generally given a dual Banach space X∗and a bounded bilinear function

F : A×A −→ X∗such that a −→ F(a,b) and b −→ F(a,b) are ω∗−ω∗-continuous,

we define?FdM ∈ X∗by

? FdM,x? =?F(a,b),x?dM(a,b)

??

(x ∈ X).

Definition 2.10. Let A be a dual Banach algebra with identity. Then a net (Mα)

in L2

ω∗(A,C)∗is called an approximate normal, virtual diagonal for A if

a.Mα− Mα.a −→ 0and∆ω∗(Mα) −→ e

From now on we assume that A is a dual Banach algebra with identity. It is well

known that every dual Banach algebra with a normal virtual diagonal is Connes-

amenable [15]. In the following theorem we extend this result to approximate

Connes-amenability.

Theorem 2.11. If A has an approximate normal, virtual diagonal {Mα}, then A is

approximately Connes-amenable.

Proof. Suppose X is a normal dual Banach A-bimodule with predual X∗ and

D ∈ Z1(A,X) ω∗− ω∗-continuous. Since A is unital, by Proposition 3.11 we can

assume that X is unital. We follow the argument in [4, Theorem 3.1]. The bilinear

map F(a,b) = Da.b is ω∗−ω∗-continuous. Thus by the above remark we may let,

?

φα=F(a,b)dMα(a,b) =

?

Da.bdMα∈ X.

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APPROXIMATE CONNES-AMENABILITY OF DUAL BANACH ALGEBRAS 9

For c ∈ A, x ∈ X∗, we have

?c.φα,x? = ?φα,x.c?

?

?

dMα(a,b) and similarly φα.c =?Da.bc

=?c.Da.b,x?dMα(a,b)

= ?c.Da.b dMα(a,b),x?.

Therefore c.φα=?c.Da.b

We have the following relations,

dMα(a,b).

?

?D(ca).b,x?dMα(a,b) = ?c.Mα,Fx? (1)

and

?

?Da.bc,x?dMα(a,b) = ?Mα.c,Fx? (2)

where Fx(a,b) = ?Da.b,x? and Fx∈ L2

ω∗(A,C).

By (1) and (2), we have

?

|?D(ca).b dMα(a,b) −

?

Da.bc dMα(a,b),x?| ≤ ?c.Mα− Mα.c??Fx?

so

?

?

Dca.bdMα(a,b) −

?

Da.bcdMα(a,b)? ≤ ?c.Mα− Mα.c??D??a??b? (3).

On the other hand since Mα=?a ⊗ bdMα(a,b), then

∆ω∗(Mα) =

?

abdMα(a,b). (4)

Thus

c.φα=

?

c.Da.bdMα(a,b)

=

?

D(ca).bdMα(a,b) −

?

Dc.abdMα(a,b).

So we have

c.φα− φα.c =

?

?

?

D(ca).bdMα(a,b)

−Da.bcdMα(a,b) −

?

Dc.abdMα(a,b)

=D(ca).bdMα(a,b) −

?

?

Da.bcdMα(a,b)

−Dc.abdMα(a,b)

By assumption and (3),(4) we have

Dc = lim

αφα.c − c.φα

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10 G. H. Esslamzadeh and B. Shojaee

= lim

αc.(−φα) − (−φα).c(c ∈ A).

Thus D is approximately inner, and hence A is approximatelyConnes-amenable.

?

Definition 2.12. A is called approximately strongly Connes-amenable if for each

unital Banach A -bimodule X, every ω∗− ω∗continuous derivation D : A −→ X∗

whose range consists of ω∗− elements is approximately inner.

We don’t know whether the converse of Theorem 2.11 is true, but for approximate

strong Connes-amenability, the corresponding question is easy to answer:

Theorem 2.13. The following conditions are equivalent:

(i) A has an approximate normal, virtual diagonal

(ii) A is approximately strongly Connes-amenable.

Proof. (i) =⇒ (ii). This is similar to Theorem 3.13.

(ii) =⇒ (i) We follow the argument in [10, Theorem 4.7]. Since ∆ω∗ is ω∗− ω∗

continuous then ker∆ω∗ is ω∗− closed and

(L2

ω∗(A,C)∗/⊥Ker∆ω∗)∗= ker∆ω∗.

Clearly ade⊗eattains its values in the ω∗− elements of ker∆ω∗. By the definition

of approximate strong Connes-amenability, there exists a net (Nα) ⊂ ker∆ω∗ such

that

ade⊗e(a) = lim

αa.Nα− Nα.a(a ∈ A).

Let Mα= e ⊗ e − Nα. It follows that

a.Mα− Mα.a −→ 0and∆ω∗(Mα) −→ e(a ∈ A).

?

One of the unsatisfactory sides of dealing with approximate Connes-amenability

for dual Banach algebras is the apparent lack of a suitable intrinsic characteriza-

tion in terms of approximate normal, virtual diagonals. we saw that dual Banach

algebras with an approximate normal, virtual diagonal are approximately Connes-

amenable, but the converse is likely to be false in general. However for a compact

group G the converse is true:

Theorem 2.14. Let G be a compact group. Then there is an approximate normal,

virtual diagonal for M(G).

Proof. By [12, Proposition 3.3] this is immediate.

?

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APPROXIMATE CONNES-AMENABILITY OF DUAL BANACH ALGEBRAS11

We now modify the definition of approximate normal, virtual diagonal and obtain

the desired characterization. Let A be a dual Banach algebra with predual A∗

and let ∆ : A? ⊗A −→ A be the multiplication map. From [12, Corollary 4.6], we

conclude that ∆∗maps A∗ into σWC((A? ⊗A)∗)[14]. Consequently, ∆∗∗induces

the homomorphism

∆σWC: σWC((A? ⊗A)∗)∗−→ A.

With these preparations made, we can now characterize approximately Connes-

amenable, dual Banach algebras through the existence of certain approximate

normal, virtual diagonals. This is indeed an approximate version [12, Theorem

4.8].

Definition 2.15. An approximate σWC− virtual diagonal for A is a net (Mα) ⊂

σWC((A? ⊗A)∗)∗such that

a.Mα− Mα.a −→ 0and∆σWC(Mα) −→ e(a ∈ A).

Theorem 2.16. The following conditions are equivalent:

(i) A is approximately Connes-amenable.

(ii)There is an approximate σWC− virtual diagonal for A.

Proof (i) =⇒ (ii): First, note that A? ⊗A is canonically mapped into σWC((A? ⊗A)∗)∗.

Define a derivation

D : A −→ σWC((A? ⊗A)∗)∗,a ?−→ a ⊗ e − e ⊗ a.

Since the dual module σWC((A? ⊗A)∗)∗is normal, then it follows that D is ω∗−continuous.

Clearly, D attains its values in the ω∗−closed submodule Ker∆σWC. Since it is a

normal dual Banach A-module, then there is a net (Nα) ⊂ ker∆σWC such that

Da = lim

αa.Nα− Nα.a(a ∈ A).

Letting Mα= e⊗e−Nα, we see that it is the approximate σWC−virtual diagonal

for A.

(ii) =⇒ (i) Let X be a normal, dual Banach A-bimodule. By Proposition 2.9 we

may assume that X is unital, and let D : A −→ X be a ω∗−derivation. Define

θD: A? ⊗A −→ X,a ⊗ b ?−→ a.Db.

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12 G. H. Esslamzadeh and B. Shojaee

By [12, lemma 4.9], θ∗

Dmaps the predual X∗into σWC((A? ⊗A)∗). Hence

(θ∗|E∗)∗: σWC((A? ⊗A)∗)∗−→ E.

Let (Mα) ⊂ σWC((A? ⊗A)∗)∗be an approximate σWC−virtual diagonal for A,

and let xα= (θ∗|E∗)∗(Mα). We have the restriction map

ı∗: (A? ⊗A)∗∗−→ σWC((A? ⊗A)∗)∗.

k=1aα

Therefore we can choose a net Σnα

k⊗ bα

k∈ A ⊗ A such that

Mα= ω∗− lim

αΣnα

k=1aα

k⊗ bα

k

on σWC((A? ⊗A)∗).

For every c ∈ A,

xα.c = ω∗− lim

αΣnα

k=1aα

k.D(bα

k).c

= ω∗− lim

α(Σnα

k=1aα

k.D(bα

kc) − Σnα

k=1aα

kbα

k.Dc)

= (θ∗|E∗)∗(Mα.c) − ∆σWC(Mα).Dc.

Therefore

c.xα− xα.c = (θ∗|E∗)∗(c.Mα− Mα.c) + ∆σWC(Mα).Dc.(1)

So by assumption and (1), we have

Dc = lim

αc.xα− xα.c(c ∈ A).

This implies that A is approximately Connes-amenable.

?

References

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Acknowledgment

The authors would like to express their sincere thanks to Professor F. Ghahramani

for his valuable comments.

G. H. Esslamzadeh

Department of Mathematics, Faculty of Sciences, Shiraz University, Shiraz 71454, Iran

e-mail: esslamz@shirazu.ac.ir

B. Shojaee

Islamic Azad University of Karadj, Karadj, Iran

e-mail: shoujaei@kiau.ac.ir

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