Approximate Connes-amenability of dual Banach algebras

0.12 MB·English

by  G. H. Esslamzadeh

APPROXIMATE CONNES-AMENABILITY OF
DUAL BANACH ALGEBRAS
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G. H. Esslamzadeh and B. Shojaee
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g Abstract. We introduce the notions of approximate Connes-amenability and
u approximatestrongConnes-amenability.Thenwecharacterizethesetwotypes
A
ofdualBanachalgebrasintermsofapproximatenormalvirtualdiagonaland
approximateσWC−virtualdiagonals.Someconcretecasesarealsodiscussed.
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MathematicsSubjectClassification(2000).Primary46H25,46H20;Secondary

46H35.
]
A
Keywords.Approximatelyinnerderivation,Approximatelyweaklyamenable,
F Approximately amenable, Approximatetrace extension property.
.
h
t
a
m
[

1 1. INTRODUCTION
v
6
6 In [9], B.E. Johnson, R. V. kadison, and J. Ringrose introduced a notion of
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3 amenability for von neuman algebras which modified Johnson’s original defini-
.
8 tion for Banach algebras[8] in the sense that it takes the dual space structure
0
9 of a von Neumann algebra into account. This notion of amenability was later
0
: called Connes-amenability by A. Ya. Helemskii [7]. In [10], the author extended
v
i
X the notionofConnes-amenabilityto the largerclassof dualBanachalgebras.The
r
a 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 versionof 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
theIslamicAzadUniversity.ThesecondauthorwishestothankIslamicAzadUniversityforthe
financialsupport.
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
δ (a)=a.x−x.a (a∈A).
x
Thenδ isaderivation;mapsofthisformarecalledinnerderivation.Aderivation
x
D : A −→ X is approximately inner if there exists a net (a ) ⊆ X such that for
α
everya∈A, D(a)=lim (a.x −x .a), the limit being in norm.We saythat A is
α α α
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 dualBanachalgebra(withA =c (G)), orif Ais anArens
∗ 0
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−→X, a7−→
(cid:26) 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.φ
A−→X∗, a7−→
(cid:26) 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.
APPROXIMATE CONNES-AMENABILITY OF DUAL BANACH ALGEBRAS3
Lemma 2.2. Suppose that A is approximately Connes-amenable. Then A has left
and right approximate identities. In particular A2 is 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=ax and x.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 (a )⊆X such that
α
a=limaa (a∈A).
α
α
This means that A has a rightapproximateidentity. Similarly,one see that A has
a left approximate identity. (cid:3)
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
∗
k(µ,α)k=max(kµk,|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 = D +ad where D : A# −→ e.X∗.e
1 η 1
is a ω∗-continuous derivation and η ∈ X∗. Since e.X∗.e is a normal dual Ba-
nach A-bimodule, then D (e) = 0 and D | is approximately inner; whence D is
1 1 A
approximately inner. Thus A# is approximately Connes-amenable.
Now suppose D :A−→X∗ is a ω∗-continuous derivation for normaldual Banach
A-bimodule X∗. Set
D :A# −→X∗, D(a+λe)=Da (a∈A,λ∈C).
e e
If we define e.x = x.e = x (e ∈ A#,x ∈ X), then X∗ turns into a normal dual
BanachA#-bimodule andD is aω∗-continuousderivation.SoD is approximately
inner,andhence soisD.ItefollowsthatAisapproximatelyConenes-amenable. (cid:3)
Proposition2.4.LetAbe anArensregulardualBanachalgebra.IfA∗∗ is approx-
imately Connes-amenable, then A is approximately Connes-amenable.
4 G. H. Esslamzadeh and B. Shojaee
Proof. Suppose X is a normal dual Banach A-bimodule, and π :A∗∗ −→A is the
restrictionmap 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 (x )⊆X such that
α
Doπ(a∗∗)=lima∗∗.x −x .a∗∗ (a∗∗ ∈A∗∗).
α α
α
So
D(a)=lima.x −x .a (a∈A).
α α
α
(cid:3)
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, Theorem3.2], G is amenable, and by [11,Theorem 4.4.13],M(G) is
approximately Connnes-amenable. (cid:3)
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, Proposition2.5] 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 BanachA-bimodule, and let D ∈Z1(A,X∗). By [11, Theo-
rem4.4.8]X∗ isanormaldualBanachA∗∗-bimoduleandDhasauniqueextension
D ∈Z1(A∗∗,X∗). From the approximate Connes-amenability of A∗∗ we conclude
tehat D, and hence D is inner. It follows that A is approximately amenable. (cid:3)
In theefollowing proposition we obtain a criterion for approximate amenability
which will be used in the sequel.
Proposition 2.7. The following conditions are equivalent;
APPROXIMATE CONNES-AMENABILITY OF DUAL BANACH ALGEBRAS5
(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∗∗)⊆X∗∗ such that
α
ι◦D(a)=lima.x∗∗−x∗∗.a (a∈A).
α α
α
Now take ǫ>0, and finite sets F ⊆A, φ⊆X∗.
Then there is α such that
|hx∗,ι◦D(a)−(a.x∗∗−x∗∗.a)i|<ǫ
α α
For all x∗ ∈φ and a∈F.
By Goldstine’s Theorem, there is a net (x ) in X such that
α
|hx∗,ι◦D(a)−(a.x −x .a)i|<ǫ
α α
for all x∗ ∈φ and a∈F. Thus there is a net (x )⊆X such that
α
Da=ω−lima.x −x .a (a∈A).
α α
α
Finally, for each finite set F ⊆A, say F ={a ,...,a },
1 n
(a .x −x .a ,...,a .x −x .a )−→(Da ,...,Da )
1 α α 1 n α α n 1 n
weakly in (X)n. By Mazure’s Theorem,
k.k
(Da ,...,Da )∈Co {(a .x −x .a ,...,a .x −x .a )},
1 n 1 α α 1 n α α n
Thus there is a convex linear combination x of elements in the set {x } such
F,ǫ α
that ,
kDa−(a.x −x .a)k<ǫ (a∈F).
(F,ǫ) (F,ǫ)
The family of such pairs (F,ǫ) is a directed set for the partial order ≤ given by
(F ,ǫ )≤(F ,ǫ ) if F ⊆F and ǫ ≥ǫ .
1 1 2 2 1 2 1 2
6 G. H. Esslamzadeh and B. Shojaee
Thus
Da= lim a.x −x .a (a∈A).
(F,ǫ) (F,ǫ)
(F,ǫ)
(cid:3)
One might ask whether the ideal condition in the Theorem2.6 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 BanachA-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 normaldual BanachA∗∗-bimodule. Let (a′′) be a net in
α
A∗∗ such that a′′ −→ω∗ a′′. Then
α
aa′′ −→ω∗ aa′′ (a∈A).
α
Since A is a right ideal of A∗∗ and ω∗-topology of A∗∗ restricted to A coincides
with the weak topology, we have
aa′′ −→ω aa′′ (a∈A).
α
Let x′′′′ ∈ X∗∗∗∗. Then by (1) there exists a ∈ A and y′′′′ ∈ X∗∗∗∗ such that
′′′′ ′′′′
x =y a. Thus we have,
x′′′′a′′ =y′′′′.aa′′ −→ω y′′′′.aa′′ =x′′′′a′′.
α α
Therefore
x′′′′a′′ −→ω∗ x′′′′a′′.
α
APPROXIMATE CONNES-AMENABILITY OF DUAL BANACH ALGEBRAS7
On the other hand by definition,
a′′x′′′′ −→ω∗ a′′x′′′′
α
Nowsince A∗∗ is approximatelyConnes-amenable,thenthere existsa net(x′′′′)∈
α
′′′′
X , such that
D∗∗a′′ =lima′′.x′′′′ −x′′′′.a′′ (a′′ ∈A∗∗).
α α
α
′′′′ ′′
Let P :X −→X be the natural projection, then
′′′′ ′′′′
Da=lima.P(x )−P(x ).a (a∈A).
α α
α
Thus D is approximately inner. It follow that A is approximately amenable. (cid:3)
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 = D +ad where
1 η
D :A−→e.X.e is a derivation and η ∈X∗. Since D is a ω∗−continuous deriva-
1
tion and X is a normal dual Banach bimodule then D is ω∗−continuous and
1
e.X.e is normal. So by assumption D is approximately inner, and therefore A is
1
approximately Connes-amenable. (cid:3)
Let A be a dual Banach algebra with identity, and let L2ω∗(A,C) be the space of
separately ω∗ −ω∗-continuous bilinear functionals on A. Clearly, L2ω∗(A,C) is a
Banach A-submodule of L2(A,C) and
L2(A,C)≃(A⊗A)∗.
b
From [10] we have a natural A-bimodule map
θ :A⊗A−→L2ω∗(A,C)∗
defined by letting θ(a⊗b)F = F(a,b). Since A∗⊗A∗ ⊆ L2ω∗(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⊆L2ω∗(A,C)∗.
8 G. H. Esslamzadeh and B. Shojaee
The map ∆ is defined as follow:
A
∆ :A⊗A−→A, a⊗b7−→ab (a,b∈A)
A
b
Since multiplication in a dual Banach algebra is separately ω∗ −ω∗-continuous,
we have,
∆∗AA∗ ⊂L2ω∗(A,C),
So that restriction, of ∆∗A∗ to L2ω∗(A,C) turns into a Banach A-bimodule homo-
morphism
∆ω∗ :L2ω∗(A,C)∗ −→A
Suppose F ∈L2ω∗(A,C) and M ∈L2ω∗(A,C)∗. We have the notation,
F(a,b)dM(a,b)= FdM :=hM,Fi.
Z Z
More generally given a dual Banach space X∗ and a bounded bilinear function
F :A×A−→X∗suchthata−→F(a,b)andb−→F(a,b)areω∗−ω∗-continuous,
we define FdM ∈X∗ by
R
h FdM,xi= hF(a,b),xidM(a,b) (x∈X).
Z Z
Definition 2.10. Let A be a dual Banach algebra with identity. Then a net (M )
α
in L2ω∗(A,C)∗ is called an approximate normal, virtual diagonal for A if
a.Mα−Mα.a−→0 and ∆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.
Theorem2.11.If A has anapproximate normal,virtualdiagonal{M }, then A is
α
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
assumethat X is unital.We followthe argumentin[4,Theorem3.1].The bilinear
mapF(a,b)=Da.b is ω∗−ω∗-continuous.Thus by the aboveremarkwe maylet,
φ = F(a,b)dM (a,b)= Da.bdM ∈X.
α α α
Z Z
APPROXIMATE CONNES-AMENABILITY OF DUAL BANACH ALGEBRAS9
For c∈A, x∈X , we have
∗
hc.φ ,xi=hφ ,x.ci
α α
= hc.Da.b,xidM (a,b)
α
Z
=h c.Da.b dM (a,b),xi.
α
Z
Therefore c.φ = c.Da.b dM (a,b) and similarly φ .c = Da.bc dM (a,b).
α α α α
R R
We have the following relations,
hD(ca).b,xidM (a,b)=hc.M ,F i (1)
α α x
Z
and
hDa.bc,xidM (a,b)=hM .c,F i (2)
α α x
Z
where Fx(a,b)=hDa.b,xi and Fx ∈L2ω∗(A,C).
By (1) and (2), we have
|h D(ca).b dM (a,b)− Da.bc dM (a,b),xi|≤kc.M −M .ckkF k
α α α α x
Z Z
so
k Dca.bdM (a,b)− Da.bcdM (a,b)k≤kc.M −M .ckkDkkakkbk (3).
α α α α
Z Z
On the other hand since M = a⊗bdM (a,b), then
α α
R
∆ω∗(Mα)= abdMα(a,b). (4)
Z
Thus
c.φ = c.Da.bdM (a,b)
α α
Z
= D(ca).bdM (a,b)− Dc.abdM (a,b).
α α
Z Z
So we have
c.φ −φ .c= D(ca).bdM (a,b)
α α α
Z
− Da.bcdM (a,b)− Dc.abdM (a,b)
α α
Z Z
= D(ca).bdM (a,b)− Da.bcdM (a,b)
α α
Z Z
−Dc. abdM (a,b)
α
Z
By assumption and (3),(4) we have
Dc=limφ .c−c.φ
α α
α
10 G. H. Esslamzadeh and B. Shojaee
=limc.(−φ )−(−φ ).c (c∈A).
α α
α
ThusDisapproximatelyinner,andhenceAisapproximatelyConnes-amenable. (cid:3)
Definition 2.12. A is called approximately strongly Connes-amenable if for each
unital BanachA -bimodule X, every ω∗−ω∗ continuous derivation D :A−→X∗
whose range consists of ω∗− elements is approximately inner.
Wedon’tknowwhethertheconverseofTheorem2.11istrue,butforapproximate
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
(L2ω∗(A,C)∗/⊥Ker∆ω∗)∗ =ker∆ω∗.
Clearly ade⊗e attains its values in the ω∗− elements of ker∆ω∗. By the definition
ofapproximatestrongConnes-amenability,thereexistsanet(Nα)⊂ker∆ω∗ such
that
ad (a)=lima.N −N .a (a∈A).
e⊗e α α
α
Let M =e⊗e−N . It follows that
α α
a.Mα−Mα.a−→0 and ∆ω∗(Mα)−→e (a∈A). (cid:3)
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
algebraswithanapproximatenormal,virtualdiagonalareapproximatelyConnes-
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. (cid:3)