connes-amenability of bidual and weighted semigroup algebras

30 Pages·2006·0.2 MB·English

MATH.SCAND.99(2006),217–246
CONNES-AMENABILITY OF BIDUAL AND
WEIGHTED SEMIGROUP ALGEBRAS
MATTHEWDAWS
Abstract
WeinvestigatethenotionofConnes-amenability,introducedbyRundein[10],forbidualalgebras
andweightedsemigroupalgebras.WeprovidesomesimplificationstothenotionofaσWC-virtual
diagonal,asintroducedin[13],especiallyinthecaseofthebidualofanArensregularBanach
algebra.Weapplytheseresultstodiscrete, weighted, weaklycancellativesemigroupalgebras,
showingthatthesebehaveinthesamewayasC∗-algebraswithregardsConnes-amenabilityof
the bidual algebra. We also show that for each one of these cancellative semigroup algebras
l1(S,ω),wehavethatl1(S,ω)isConnes-amenable(withrespecttothecanonicalpredualc0(S))
ifandonlyifl1(S,ω)isamenable,whichisinturnequivalenttoSbeinganamenablegroup,and
theweightsatisfyingacertainrestrictivecondition.ThislatterpointwasfirstshownbyGrønbæk
in[6],butweprovideaunifiedproof.Finally,weconsiderthehomologicalnotionofinjectivity,
andshowthathere,weightedsemigroupalgebrasdonotbehavelikeC∗-algebras.
1. Introduction
Wefirstfixsomenotation,following[2].ForaBanachspaceE,weletE(cid:2)beits
dualspace,andforµ∈E(cid:2) andx ∈E,wewrite(cid:4)µ,x(cid:5)=µ(x)fornotational
convenience. We then have the canonical map κE : E → E(cid:2)(cid:2) defined by
(cid:4)κE(x),µ(cid:5) = (cid:4)µ,x(cid:5) for µ ∈ E(cid:2),x ∈ E. For Banach spaces E and F, we
writeB(E,F)fortheBanachspaceofboundedlinearmapsbetweenE and
F;wewriteB(E,E) = B(E);wewriteT(cid:2) fortheadjointofT ∈ B(E,F).
WeusethenotionofBanachleftA-modules,rightmodulesandbimodulesas
in[2].
A linear map d : A → E between a Banach algebra A and a Banach
A-bimodule E is a derivation if d(ab) = a ·d(b)+d(a)·b for a,b ∈ A.
For x ∈ E, we define δx : A → E by δx(a) = a ·x −x ·a. Then δx is a
derivation,calledaninnerderivation.
A Banach algebra A is amenable if every derivation d : A → E(cid:2) to a
dualbimoduleisinner.Forexample,aC∗-algebraA isamenableifandonly
ifA isnuclear; agroupalgebraL1(G)isamenableifandonlyifthelocally
compactgroupGisamenable(whichisthemotivatingexample).See[14]for
furtherdiscussionsofamenabilityandrelatednotions.
ReceivedSeptember5,2005.
218 matthewdaws
LetEbeaBanachspaceandF aclosedsubspaceofE.Thenwenaturally,
isometrically,identifyF(cid:2) withE(cid:2)/F◦,where
F◦ ={µ∈E(cid:2) :(cid:4)µ,x(cid:5)=0 (x ∈F)}.
Definition1.1. LetEbeaBanachspaceandE∗beaclosedsubspaceof
E(cid:2). Let πE∗ : E(cid:2)(cid:2) → E(cid:2)(cid:2)/E∗◦ be the quotient map, and suppose that πE∗ ◦κE
isanisomorphismfromE toE(cid:2).ThenwesaythatE isadualBanachspace
∗
withpredualE∗.
WhenA isadualBanachspacewithpredualA∗whichisalsoasubmodule
ofA(cid:2) wesaythatA isadualBanachalgebra.
ForadualBanachalgebraA withpredualA∗, wehenceforthidentifyA
withA∗(cid:2).Thuswegetaweak∗-topologyonA,whichwedenotebyσ(A,A∗).
AsnoticedbyRunde(see[10]),thereareveryfewBanachalgebraswhich
arebothdualandamenable.ForvonNeumannalgebras,whicharethemotiv-
atingexampleofdualBanachalgebras,thereisaweakernotionofamenablity,
called Connes-amenability, which has a natural generalisation to the case of
dualBanachalgebras.
Definition1.2. LetA beadualBanachalgebrawithpredualA∗.LetE
be a Banach A-bimodule.Then E(cid:2) is a w∗-Banach A-bimodule if, for each
µ∈E(cid:2),themaps (cid:1)
a·µ,
A →E(cid:2), a (cid:14)→
µ·a
areσ(A,A∗)−σ(E(cid:2),E)continuous.
Then (A,A∗) is Connes-amenable if, for each w∗-Banach A-bimodule
E(cid:2),eachderivationd :A →E(cid:2),whichisσ(A,A∗)−σ(E(cid:2),E)continuous,
isinner.
GivenaBanachalgebraA,wedefinebilinearmapsA(cid:2)(cid:2)×A(cid:2) → A(cid:2) and
A(cid:2)×A(cid:2)(cid:2) →A(cid:2) by
(cid:4)(cid:26)·µ,a(cid:5)=(cid:4)(cid:26),µ·a(cid:5) (cid:4)µ·(cid:26),a(cid:5)=(cid:4)(cid:26),a·µ(cid:5) ((cid:26)∈A(cid:2)(cid:2),µ∈A(cid:2),a ∈A).
Wethendefinetwobilinearmaps(cid:1),♦:A(cid:2)(cid:2)×A(cid:2)(cid:2) →A(cid:2)(cid:2) by
(cid:4)(cid:26)(cid:1)(cid:27),µ(cid:5)=(cid:4)(cid:26),(cid:27)·µ(cid:5) (cid:4)(cid:26)♦(cid:27),µ(cid:5)=(cid:4)(cid:27),µ·(cid:26)(cid:5) ((cid:26),(cid:27) ∈A(cid:2)(cid:2),µ∈A(cid:2)).
Wecancheckthat(cid:1)and♦areactuallyalgebraproducts,calledthefirst and
secondArensproductsrespectively.ThenκA :A →A(cid:2)(cid:2) isahomomorphism
with respect to eitherArens product.When (cid:1) = ♦, we say that A is Arens
regular. In particular, when A isArens regular, we may check that A(cid:2)(cid:2) is a
dualBanachalgebrawithpredualA(cid:2).
connes-amenabilityofbidualandweightedsemigroup... 219
Theorem 1.3. Let A be an Arens regular Banach algebra. When A is
amenable, A(cid:2)(cid:2) is Connes-amenable. If κA(A) is an ideal in A(cid:2)(cid:2) and A(cid:2)(cid:2) is
Connes-amenable,thenA isamenable.
Let A be a C∗-algebra. Then A is Arens regular, and A(cid:2)(cid:2) is Connes-
amenableifandonlyifA isamenable.
Proof. Thefirststatementsare[10,Corollary4.3]and[10,Theorem4.4].
ThestatementaboutC∗-algebrasisdetailedin[14,Chapter6].
AnotherclassofConnes-amenabledualBanachalgebrasisgivenbyRunde
in[11],whereitisshownthatM(G),themeasurealgebraofalocallycompact
groupG,isamenableifandonlyifGisamenable.
The organisation of this paper is as follows. Firstly, we study intrinsic
characterisations of amenability, recalling a result of Runde from [13]. We
then simplify these conditions in the case ofArens regular Banach algebras.
We recall the notion of an injective module, and quickly note how Connes-
amenabilitycanbephrasedinthislanguage.Thefinalsectionofthepaperthen
appliestheseideastoweightedsemigroupalgebras.Wefinishwithsomeopen
questions.
2. Characterisations of amenability
LetEandF beBanachspaces,andformthealgebraictensorproductE⊗F.
WecannormE⊗F withtheprojectivetensornorm,definedas
(cid:1) (cid:3)
(cid:2)n (cid:2)n
(cid:17)u(cid:17)π =inf (cid:17)xk(cid:17)(cid:17)yk(cid:17):u= xk ⊗yk (u∈E⊗F).
k=1 k=1
Thenthecompletionof(E⊗F,(cid:17)·(cid:17)π)isE⊗(cid:4)F,theprojectivetensorproduct
ofE andF.
LetA beaBanachalgebra.ThenA ⊗(cid:4)A isaBanachA-bimoduleforthe
moduleactionsgivenby
a·(b⊗c)=ab⊗c, (b⊗c)·a =b⊗ca (a ∈A,b⊗c ∈A ⊗(cid:4)A).
Define"A :A⊗(cid:4)A →A by"A(a⊗b)=ab.Then"A isanA-bimodule
homomorphism.LetM ∈(A ⊗(cid:4)A)(cid:2)(cid:2) besuchthat
a·M =M ·a, "(cid:2)(cid:2) (M)·a =κ (a) (a ∈A).
A A
The M is a virtual diagonal for A. It is well-known that A is an amenable
BanachalgebraifandonlyifA hasavirtualdiagonal.
220 matthewdaws
Definition 2.1. Let A be a dual Banach algebra with predual A∗, and
let E be a Banach A-bimodule.Then x ∈ σWC(E) if and only if the maps
A →E, (cid:5)a·x,
a (cid:14)→
x ·a
areσ(A,A∗)−σ(E,E(cid:2))continuous.
It is clear that σWC(E) is a closed submodule of E. The A-bimodule
homomorphism"A hasadjoint"(cid:2)A :A(cid:2) →(A⊗(cid:4)A)(cid:2).In[13,Corollary4.6]
itisshownthat"(cid:2)A(A∗)⊆σWC((A⊗(cid:4)A)(cid:2)).Consequently,wecanview"(cid:2)A
asamapA∗ →σWC((A⊗(cid:4)A)(cid:2)),andhenceview"(cid:2)A(cid:2) asamapσWC((A⊗(cid:4)
A)(cid:2))(cid:2) →A∗(cid:2) =A,denotedby"˜A.LetM ∈σWC((A ⊗(cid:4)A)(cid:2))(cid:2)besuchthat
a·M =M ·a, a"˜ (M)=a (a ∈A).
A
TheM isaσWC-virtualdiagonalforA.
Theorem2.2. LetA beadualBanachalgebrawithpredualA∗.Thenthe
followingareequivalent:
(1) A isConnes-amenable;
(2) A hasaσWC-virtualdiagonal.
Proof. Thisis[13,Theorem4.8].
In particular, we see that a Connes-amenable Banach algebra is unital
(which can of course be shown in an elementary fashion, as in [10, Proposi-
tion4.1]).
3. Connes-amenability for biduals of algebras
Recall Gantmacher’s theorem, which states that a bounded linear map T :
E → F between Banach spaces E and F is weakly compact if and only if
T(cid:2)(cid:2)(E(cid:2)(cid:2)) ⊆ κF(F).We write W(E,F) for the collection of weakly compact
operatorsinB(E,F).
Lemma 3.1. Let E be a dual Banach space with predual E∗, let F be a
Banachspace,andletT ∈B(E,F(cid:2)).Thenthefollowingareequivalent,and
inparticulareachimplythatT isweaklycompact:
(1) T isσ(E,E∗)−σ(F(cid:2),F(cid:2)(cid:2))continuous;
(2) T(cid:2)(F(cid:2)(cid:2))⊆κE (E∗);
∗
(3) thereexistsS ∈W(F,E∗)suchthatS(cid:2) =T.
Proof. That(1)and(2)areequivalentisstandard.
connes-amenabilityofbidualandweightedsemigroup... 221
Supposethat(2)holds,sothatwemaydefineS ∈B(F,E∗)byκE ◦S =
∗
T(cid:2)◦κF.Then,forx ∈E andy ∈F,wehave
(cid:4)x,S(y)(cid:5)=(cid:4)T(cid:2)(κ (y)),x(cid:5)=(cid:4)T(x),y(cid:5),
F
so that S(cid:2) = T. Then S(cid:2)(cid:2)(F(cid:2)(cid:2)) = T(cid:2)(F(cid:2)(cid:2)) ⊆ κE (E∗), so that S is weakly
∗
compact,byGantmacher’sTheorem,sothat(3)holds.
Conversely, if (3) holds, as S is weakly compact, we have κE∗(E∗) ⊇
S(cid:2)(cid:2)(F(cid:2)(cid:2))=T(cid:2)(F(cid:2)(cid:2)),sothat(2)holds.
ItisstandardthatforBanachspacesEandF,wehave(E⊗(cid:4)F)(cid:2) =B(F,E(cid:2))
withdualitydefinedby
(cid:4)T,x ⊗y(cid:5)=(cid:4)T(y),x(cid:5) (T ∈B(F,E(cid:2)),x ⊗y ∈E⊗(cid:4)F).
Then we see, for a,b,c ∈ A and T ∈ (A ⊗(cid:4) A)(cid:2) = B(A,A(cid:2)), that (cid:4)a ·
T,b⊗c(cid:5) = (cid:4)T(ca),b(cid:5)andthat(cid:4)T ·a,b⊗c(cid:5) = (cid:4)T(c),ab(cid:5) = (cid:4)T(c)·a,b(cid:5)
sothat
(1)
(a·T)(c)=T(ca), (T ·a)(c)=T(c)·a (a,c ∈A,T :A →A(cid:2)).
Notice that we could also have defined (E ⊗(cid:4) F)(cid:2) to be B(E,F(cid:2)). This
wouldinduceadifferentbimodulestructureonB(A,A(cid:2)),butweshallseein
Section4thatourchosenconventionseemsmorenaturalforthetaskathand.
Proposition3.2. LetA beadualBanachalgebrawithpredualA∗.For
T ∈B(A,A(cid:2))=(A ⊗(cid:4)A)(cid:2),definemapsφr,φl :A ⊗(cid:4)A →A(cid:2) by
φ (a⊗b)=T(cid:2)κ (a)·b, φ(a⊗b)=a·T(b) (a⊗b ∈A ⊗(cid:4)A).
r A l
ThenT ∈σWC(B(A,A(cid:2)))ifandonlyifφr andφl areweaklycompactand
haverangescontainedinκA (A∗).
∗
Proof. For T ∈ B(A,A(cid:2)) = (A ⊗(cid:4)A)(cid:2), define RT,LT : A → (A ⊗(cid:4)
A)(cid:2) by RT(a) = a · T and LT = T · a, for a ∈ A. By definition, T ∈
σWC(B(A,A(cid:2)))ifandonlyifRT andLT areσ(A,A∗)−σ(B(A,A(cid:2)),
(A⊗(cid:4)A)(cid:2)(cid:2))continuous.ByLemma3.1,thisisifandonlyifthereexistϕr,ϕl ∈
W(A ⊗(cid:4)A,A∗)suchthatϕr(cid:2) =RT andϕl(cid:2) =LT.
Fora⊗b ∈A ⊗(cid:4)A andc ∈A,weseethat
(cid:4)c,ϕ (a⊗b)(cid:5)=(cid:4)R (c),a⊗b(cid:5)=(cid:4)c·T,a⊗b(cid:5)=(cid:4)T(bc),a(cid:5)
r T
=(cid:4)T(cid:2)κ (a),bc(cid:5)=(cid:4)T(cid:2)κ (a)·b,c(cid:5)=(cid:4)φ (a⊗b),c(cid:5),
A A r
(cid:4)c,ϕ(a⊗b)(cid:5)=(cid:4)L (c),a⊗b(cid:5)=(cid:4)T ·c,a⊗b(cid:5)=(cid:4)T(b),ca(cid:5)
l T
=(cid:4)a·T(b),c(cid:5)=(cid:4)φ(a⊗b),c(cid:5).
l
222 matthewdaws
Thus κA ◦ ϕr = φr and κA ◦ ϕl = φl. Consequently, we see that T ∈
∗ ∗
σWC(B(A,A(cid:2)))ifandonlyifφr andφlareweaklycompactandtakevalues
inκA (A∗).
∗
Thefollowingdefinitionis[13,Definition4.1].
Definition 3.3. Let A be a Banach algebra and let E be a Banach A-
bimodule.Anelementx ∈E isweaklyalmostperiodicifthemaps
(cid:5)
a·x,
A →E, a (cid:14)→
x ·a
areweaklycompact.ThecollectionofweaklyalmostperiodicelementsinE
isdenotedbyWAP(E).
Lemma 3.4. Let A be a Banach algebra, and let T ∈ B(A,A(cid:2)) =
(A⊗(cid:4)A)(cid:2).Letφr,φl :A⊗(cid:4)A →A(cid:2)beasabove.ThenT ∈WAP(B(A,A(cid:2)))
ifandonlyifφr andφl areweaklycompact.
Proof. LetRT,LT :A →B(A,A(cid:2))beasintheaboveproof.Bydefin-
ition, T ∈ WAP(B(A,A(cid:2))) if and only if LT and RT are weakly compact.
Wecanverifythat
φ(cid:2) ◦κ =R , φ(cid:2)◦κ =L , R(cid:2) ◦κ =φ , L(cid:2) ◦κ =φ,
r A T l A T T A⊗(cid:4)A r T A⊗(cid:4)A l
whichcompletestheproof.
Corollary3.5. LetA beaunital,dualBanachalgebrawithpredualA∗,
and let T ∈ B(A,A(cid:2)) = (A ⊗(cid:4) A)(cid:2).The following are equivalent, and, in
particular,eachimplythatT isweaklycompact:
(1) T ∈σWC(B(A,A(cid:2)));
(2) T(A)⊆κA (A∗),T(cid:2)(κA(A))⊆κA (A∗),andT∈σWC(B(A,A(cid:2)));
∗ ∗
(3) T(A)⊆κA (A∗),T(cid:2)(κA(A))⊆κA (A∗),andT∈WAP(B(A,A(cid:2))).
∗ ∗
Proof. Let eA be the unit of A, so that for a ∈ A, we have T(a) =
φl(eA ⊗a) and T(cid:2)κA(a) = φr(a ⊗eA), which shows that (1) implies (2);
clearly(2)implies(1).
As A∗ is an A-bimodule, (2) and (3) are equivalent by an application of
Lemma3.4andProposition3.2.
Theorem3.6. LetA beadualBanachalgebrawithpredualA∗.ThenA
isConnes-amenableifandonlyifA isunitalandthereexistsM ∈(A⊗(cid:4)A)(cid:2)(cid:2)
suchthat:
(1) (cid:4)M,a·T −T ·a(cid:5)=0fora ∈A andT ∈σWC(W(A,A(cid:2)));
(2) κA(cid:2) "(cid:2)A(cid:2) (M)=eA,whereeA istheunitofA.
∗
connes-amenabilityofbidualandweightedsemigroup... 223
Proof. As σWC((A ⊗(cid:4) A)(cid:2))(cid:2) is a quotient of (A ⊗(cid:4) A)(cid:2)(cid:2), this is just a
re-statementofTheorem2.2.
When A is an Arens regular Banach algebra, A(cid:2)(cid:2) is a dual Banach al-
gebrawithcanonicalpredualA(cid:2).Inthiscase,wecanmakesomesignificant
simplificationsinthecharacterisationofwhenA(cid:2)(cid:2) isConnes-amenable.
ForaBanachalgebraA,wedefinethemapκA⊗κA :A⊗(cid:4)A →A(cid:2)(cid:2)⊗(cid:4)A(cid:2)(cid:2)
by
(κ ⊗κ )(a⊗b)=κ (a)⊗κ (b) (a⊗b ∈A ⊗(cid:4)A).
A A A A
WeturnA(cid:2)(cid:2)⊗(cid:4)A(cid:2)(cid:2)intoaBanachA-bimoduleinthecanonicalway.ThenκA⊗
κA isanA-bimodulehomomorphism.Thefollowingisasimpleverification.
Lemma3.7. LetA beaBanachalgebra.Themap
ιA :B(A,A(cid:2))→B(A(cid:2)(cid:2),A(cid:2)(cid:2)(cid:2)); T (cid:14)→T(cid:2)(cid:2),
is an A-bimodule homomorphism which is an isometry onto its range. Fur-
thermore,wehavethat(κA⊗κA)(cid:2)◦ιA =IB(A,A(cid:2)).DefineρA :A(cid:2)(cid:2)⊗(cid:4)A(cid:2)(cid:2) →
(A ⊗(cid:4)A)(cid:2)(cid:2) by
(cid:4)ρ (τ),T(cid:5)=(cid:4)T(cid:2)(cid:2),τ(cid:5) (τ ∈A(cid:2)(cid:2)⊗(cid:4)A(cid:2)(cid:2),T ∈B(A,A(cid:2))=(A ⊗(cid:4)A)(cid:2)).
A
Then ρA is a norm-decreasing A-bimodule homomorphism which satisfies
ρA ◦(κA ⊗κA)=κA⊗(cid:4)A.
For a Banach algebra A, it is clear that W(A,A(cid:2)) is a sub-A-bimodule
ofB(A,A(cid:2))=(A ⊗(cid:4)A)(cid:2).
Theorem 3.8. Let A be anArens regular Banach algebra such that A(cid:2)(cid:2)
is unital, and let T ∈ B(A(cid:2)(cid:2),A(cid:2)(cid:2)(cid:2)) = (A(cid:2)(cid:2) ⊗(cid:4) A(cid:2)(cid:2))(cid:2). Then the following are
equivalent:
(1) T ∈ σWC(B(A(cid:2)(cid:2),A(cid:2)(cid:2)(cid:2))), where we treat B(A(cid:2)(cid:2),A(cid:2)(cid:2)(cid:2)) as an A(cid:2)(cid:2)-
bimodule;
(2) T=S(cid:2)(cid:2)forsomeS∈WAP(W(A,A(cid:2))),wherenowwetreatW(A,A(cid:2))
asanA-bimodule.
Proof. WeapplyCorollary3.5toA(cid:2)(cid:2),sothat(1)isequivalenttoT being
weakly compact, T(A(cid:2)(cid:2)) ⊆ κA(cid:2)(A(cid:2)), T(cid:2)(κA(cid:2)(cid:2)(A(cid:2)(cid:2))) ⊆ κA(cid:2)(A(cid:2)), and T ∈
WAP(B(A(cid:2)(cid:2),A(cid:2)(cid:2)(cid:2))). Thus, if (1) holds, then there exists T ∈ W(A(cid:2)(cid:2),A(cid:2))
0
suchthatT =κA(cid:2)◦T0,andthereexistsT1 ∈W(A(cid:2)(cid:2),A(cid:2))suchthatT(cid:2)◦κA(cid:2)(cid:2) =
κA(cid:2)◦T1.LetS =T0◦κA ∈W(A,A(cid:2)).Asbefore,wecancheckthatS(cid:2) =T1
and that S(cid:2)(cid:2) = T. We know that the maps LT,RT : A(cid:2)(cid:2) → B(A(cid:2)(cid:2),A(cid:2)(cid:2)(cid:2)),
224 matthewdaws
defined by LT((cid:26)) = T ·(cid:26) and RT((cid:26)) = (cid:26)·T for (cid:26) ∈ A(cid:2)(cid:2), are weakly
compact. Define LS,RS : A → B(A,A(cid:2)) is an analogous manner, using
S ∈W(A,A(cid:2)).Fora ∈A,S ·a ∈W(A,A(cid:2)),sofor(cid:27) ∈A(cid:2)(cid:2) andb ∈A,
(cid:4)(S ·a)(cid:2)((cid:27)),b(cid:5)=(cid:4)(cid:27),(S ·a)(b)(cid:5)=(cid:4)(cid:27),S(b)·a(cid:5)
=(cid:4)a·(cid:27),S(b)(cid:5)=(cid:4)S(cid:2)(a·(cid:27)),b(cid:5).
Thus,fora ∈A and(cid:26),(cid:27) ∈A(cid:2)(cid:2),wehavethat
(cid:4)ι (L (a))((cid:26)),(cid:27)(cid:5)=(cid:4)(S ·a)(cid:2)(cid:2)((cid:26)),(cid:27)(cid:5)=(cid:4)(cid:26),S(cid:2)(a·(cid:27))(cid:5)=(cid:4)S(cid:2)(cid:2)((cid:26))·a,(cid:27)(cid:5),
A S
sothatιA(LS(a))((cid:26))=S(cid:2)(cid:2)((cid:26))·a,andhencethatιA(LS(a))=S(cid:2)(cid:2)·a =T·a =
T ·κA(a)=LT(κA(a)).ThuswehavethatLS =(κA ⊗κA)(cid:2)◦RT ◦κA,so
thatLS isweaklycompact.AsimilarcalculationshowsthatRS isalsoweakly
compact,sothatS ∈WAP(W(A,A(cid:2))).Thisshowsthat(1)implies(2).
Conversely,if(2)holds,thenLSandRSareweaklycompact.AsSisweakly
compact, T(A(cid:2)(cid:2)) = S(cid:2)(cid:2)(A(cid:2)(cid:2)) ⊆ κA(cid:2)(A(cid:2))andT(cid:2)(κA(cid:2)(cid:2)(A(cid:2)(cid:2))) = S(cid:2)(cid:2)(cid:2)(κA(cid:2)(cid:2)(A(cid:2)(cid:2)))
= κA(cid:2)(S(cid:2)(A(cid:2)(cid:2))) ⊆ κA(cid:2)(A(cid:2)),andT isweaklycompact.Thus,toshow(1),we
arerequiredtoshowthatLT andRT areweaklycompact.
Fora,b ∈A and(cid:26)∈A(cid:2),wehave
(cid:4)(a·S)(cid:2)((cid:26)),b(cid:5)=(cid:4)(cid:26),S(ba)(cid:5)=(cid:4)a·S(cid:2)((cid:26)),b(cid:5).
Then,for(cid:26),(cid:27) ∈A(cid:2)(cid:2) anda ∈A,wethushave
(cid:4)R(cid:2)(ρ ((cid:26)⊗(cid:27))),a(cid:5)=(cid:4)(a·S)(cid:2)(cid:2),(cid:26)⊗(cid:27)(cid:5)=(cid:4)(a·S)(cid:2)(cid:2)((cid:27)),(cid:26)(cid:5)
S A
=(cid:4)(cid:27),a·S(cid:2)((cid:26))(cid:5)=(cid:4)(cid:27) ·a,S(cid:2)((cid:26))(cid:5)
=(cid:4)(cid:27) (cid:1)κ (a),S(cid:2)((cid:26))(cid:5)=(cid:4)S(cid:2)((cid:26))·(cid:27),a(cid:5).
A
HenceweseethatRS(cid:2)(ρA((cid:26)⊗(cid:27))) = S(cid:2)((cid:26))·(cid:27).LetU = RS(cid:2) ◦ρA : A(cid:2)(cid:2) ⊗(cid:4)
A(cid:2)(cid:2) →A(cid:2),sothatasRS isweaklycompact,soisU.Then,for(cid:26),(cid:27),- ∈A(cid:2)(cid:2),
wehavethat
(cid:4)U(cid:2)(-),(cid:26)⊗(cid:27)(cid:5)=(cid:4)-,S(cid:2)((cid:26))·(cid:27)(cid:5)=(cid:4)(cid:27) ♦-,S(cid:2)((cid:26))(cid:5)
=(cid:4)S(cid:2)(cid:2)((cid:27) (cid:1)-),(cid:26)(cid:5)=(cid:4)(-·S(cid:2)(cid:2))((cid:27)),(cid:26)(cid:5),
so that U(cid:2)(-) = - · T, that is, U(cid:2) = RT, so that RT is weakly compact.
Similarly,wecanshowthatLT isweaklycompact,completingtheproof.
Theorem 3.9. Let A be an Arens regular Banach algebra. Then A(cid:2)(cid:2) is
Connes-amenableifandonlyifA(cid:2)(cid:2) isunitalandthereexistsM ∈(A ⊗(cid:4)A)(cid:2)(cid:2)
suchthat:
connes-amenabilityofbidualandweightedsemigroup... 225
(1) "(cid:2)A(cid:2) (M)=eA(cid:2)(cid:2),theunitofA(cid:2)(cid:2);
(2) (cid:4)M,a·T −T ·a(cid:5)=0foreacha ∈A andeachT ∈WAP(W(A,A(cid:2))).
Proof. ByTheorem3.6,wewishtoshowthattheexistenceofsuchanM
isequivalenttotheexistenceofN ∈(A(cid:2)(cid:2)⊗(cid:4)A(cid:2)(cid:2))(cid:2)(cid:2) suchthat:
(N1) κA(cid:2) (cid:2)"(cid:2)A(cid:2) (cid:2)(cid:2)(N)=eA(cid:2)(cid:2);
(N2) (cid:4)N,(cid:26)·S −S ·(cid:26)(cid:5)=0foreach(cid:26)∈A(cid:2)(cid:2) andeach
S ∈σWC(B(A(cid:2)(cid:2),A(cid:2)(cid:2)(cid:2))).
We can verify that ιA ◦ "(cid:2)A = "(cid:2)A(cid:2)(cid:2) ◦ κA(cid:2), so that (N1) is equivalent to
"(cid:2)A(cid:2) ι(cid:2)A(N) = eA(cid:2)(cid:2). For S ∈ σWC(B(A(cid:2)(cid:2),A(cid:2)(cid:2)(cid:2))), we know that S = T(cid:2)(cid:2) for
some T ∈ WAP(W(A,A(cid:2))), byTheorem 3.8.That is, the maps φr and φl,
formedusingT asinProposition3.2,areweaklycompact.Then,for(cid:26)∈A(cid:2)(cid:2),
φ(cid:2)((cid:26)),φ(cid:2)((cid:26))∈B(A,A(cid:2)),andwecancheckthat
r l
φ(cid:2)((cid:26))(a)=κ(cid:2) T(cid:2)(cid:2)(a·(cid:26)), φ(cid:2)((cid:26))(a)=T(a)·(cid:26) (a ∈A).
r A l
Thenφ(cid:2)((cid:26))(cid:2),φ(cid:2)((cid:26))(cid:2) ∈B(A(cid:2)(cid:2),A(cid:2))arethemaps
r l
φ(cid:2)((cid:26))(cid:2)((cid:27))=(cid:26)·T(cid:2)((cid:27)), φ(cid:2)((cid:26))(cid:2)((cid:27))=T(cid:2)((cid:26)(cid:1)(cid:27)) ((cid:27) ∈A(cid:2)(cid:2)),
r l
wherewerememberthatT(cid:2)(cid:2)(A(cid:2)(cid:2)) ⊆ κA(cid:2)(A(cid:2)).Consequentlyφr(cid:2)((cid:26))(cid:2)(cid:2),φl(cid:2)((cid:26))(cid:2)(cid:2)
∈B(A(cid:2)(cid:2),A(cid:2)(cid:2)(cid:2))aregivenby
φ(cid:2)((cid:26))(cid:2)(cid:2)((cid:27))=T(cid:2)(cid:2)((cid:27) (cid:1)(cid:26)), φ(cid:2)((cid:26))(cid:2)(cid:2)((cid:27))=T(cid:2)(cid:2)((cid:27))·(cid:26) ((cid:27) ∈A(cid:2)(cid:2)),
r l
whereA(cid:2)(cid:2)(cid:2)isanA(cid:2)(cid:2)-bimodule,asA(cid:2)(cid:2)isArensregular.Thatis,φ(cid:2)((cid:26))(cid:2)(cid:2) =(cid:26)·S
r
andφ(cid:2)((cid:26))(cid:2)(cid:2) =S ·(cid:26).Hence(N2)isequivalentto
l
0 =(cid:4)N,φr(cid:2)((cid:26))(cid:2)(cid:2)−φl(cid:2)((cid:26))(cid:2)(cid:2)(cid:5)=(cid:4)N,ιA(φr(cid:2)((cid:26))−φl(cid:2)((cid:26)))(cid:5)
=(cid:4)ι(cid:2) (N),φ(cid:2)((cid:26))−φ(cid:2)((cid:26))(cid:5),
A r l
foreach(cid:26)∈A(cid:2)(cid:2) andS ∈σWC(B(A(cid:2)(cid:2),A(cid:2)(cid:2)(cid:2))).Thatis,(N2)isequivalentto
φ(cid:2)(cid:2)ι(cid:2) (N)−φ(cid:2)(cid:2)ι(cid:2) (N)=0 (S ∈σWC(B(A(cid:2)(cid:2),A(cid:2)(cid:2)(cid:2)))).
r A l A
As φr and φl are weakly compact, φr(cid:2)(cid:2) and φl(cid:2)(cid:2) take values in κA(cid:2)(A(cid:2)), and
so(N2)isequivalentto
0 =(cid:4)φr(cid:2)(cid:2)ι(cid:2)A(N)−φl(cid:2)(cid:2)ι(cid:2)A(N),κA(a)(cid:5)=(cid:4)ι(cid:2)A(N),φr(cid:2)(κA(a))−φl(cid:2)(κA(a))(cid:5),
for each a ∈ A and each S ∈ σWC(B(A(cid:2)(cid:2),A(cid:2)(cid:2)(cid:2))). However, φr(cid:2)(κA(a))−
φl(cid:2)(κA(a))=a·T −T ·a,sothat(N2)isequivalentto
0 =(cid:4)ι(cid:2) (N),a·T −T ·a(cid:5) (a ∈A),
A
226 matthewdaws
foreachT ∈W(A,A(cid:2))suchthatφr andφl areweaklycompact.
Thuswehaveestablishedthat(N1)holdsforN ifandonlyif(1)holdsfor
M =ι(cid:2) (N),andthat(N2)holdsforNifandonlyif(2)holdsforM =ι(cid:2) (N),
A A
completingtheproof.
WeimmediatelyseethatA amenableimpliesthatA(cid:2)(cid:2)isConnes-amenable.
Furthermore, if A is itself a dual Banach algebra, then Corollary 3.5 shows
thatifA(cid:2)(cid:2)isConnes-amenable,thenA isConnes-amenable:noticethatifeA(cid:2)(cid:2)
istheunitofA(cid:2)(cid:2),then
(cid:4)κ(cid:2) (e )a,µ(cid:5)=(cid:4)e ·a,κ (µ)(cid:5)
A∗ A(cid:2)(cid:2) A(cid:2)(cid:2) A∗
=(cid:4)κ (a),κ (µ)(cid:5)=(cid:4)a,µ(cid:5) (a ∈A,µ∈A ),
A A ∗
∗
sothatκA(cid:2) (eA(cid:2)(cid:2))istheunitofA.
∗
4. Injectivity of the predual module
LetA beaBanachalgebra,andletE andF beBanachleftA-modules.We
write AB(E,F) for the closed subspace of B(E,F) consisting of left A-
modulehomomorphisms,andsimilarlywriteBA(E,F)andABA(E,F)for
rightA-moduleandA-bimodulehomomorphisms,respectively.Wesaythat
T ∈ AB(E,F) is admissible if both the kernel and image of T are closed,
complemented subspaces of, respectively, E and F. If T is injective, this is
equivalenttotheexistenceofS ∈B(F,E)suchthatST =IE.
Definition4.1. LetA beaBanachalgebra,andletEbeaBanachleftA-
module.ThenEisinjectiveif,wheneverF andGareBanachleftA-modules,
θ ∈ AB(F,G)isinjectiveandadmissible, andσ ∈ AB(F,E), thereexists
ρ ∈AB(G,E)withρ ◦θ =σ.
WesaythatE isleft-injectivewhenwewishtostressthatwearetreating
E asaleftmodule.Similardefinitionsholdforrightmodulesandbimodules
(writtenright-injectiveandbi-injectivewherenecessary).
Let A be a Banach algebra, let E be a Banach left A-module, and turn
B(A,E)intoaleftA-modulebysetting
(a·T)(b)=T(ba) (a,b ∈A,T ∈B(A,E)).
ThenthereisacanonicalleftA-modulehomomorphismι : E → B(A,E)
givenby
ι(x)(a)=a·x (a ∈A,x ∈E).
Notice that if E is a closed submodule of A(cid:2), then B(A,E) is a closed
submoduleof(A⊗(cid:4)A)(cid:2) = B(A,A(cid:2)),andιistherestrictionof"(cid:2) : A(cid:2) →
A
B(A,A(cid:2))toE.