DUAL BANACH ALGEBRAS: CONNES-AMENABILITY, NORMAL, VIRTUAL

21 Pages·2004·0.14 MB·English

MATH.SCAND.95(2004),124–144
DUAL BANACH ALGEBRAS: CONNES-AMENABILITY,
NORMAL, VIRTUAL DIAGONALS, AND INJECTIVITY
OF THE PREDUAL BIMODULE
VOLKERRUNDE∗
Abstract
Let(cid:1)beadualBanachalgebrawithpredual(cid:1)∗ andconsiderthefollowingassertions:(A)(cid:1)
isConnes-amenable; (B)(cid:1)hasanormal,virtualdiagonal; (C)(cid:1)∗ isaninjective(cid:1)-bimodule.
Forgeneral(cid:1),allthatisknownisthat(B)implies(A)whereas,forvonNeumannalgebras,(A),
(B),and(C)areequivalent.Weshowthat(C)alwaysimplies(B)whereastheconverseisfalse
for(cid:1) = M(G)whereGisaninfinite,locallycompactgroup.Furthermore,wepresentpartial
solutionstowardsacharacterizationof(A)and(B)for(cid:1)=B(G)intermsofG:Foramenable,
discreteGaswellasforcertaincompactG,theyareequivalenttoGhavinganabeliansubgroup
offiniteindex.Thequestionofwhetherornot(A)and(B)arealwaysequivalentremainsopen.
However,weintroduceamodifieddefinitionofanormal,virtualdiagonaland,usingthismodified
definition,characterizetheConnes-amenable,dualBanachalgebrasthroughtheexistenceofan
appropriatenotionofvirtualdiagonal.
Introduction
In [15], B. E. Johnson, R. V. Kadison, and J. Ringrose introduced a notion
ofamenabilityforvonNeumannalgebraswhichmodifiesJohnson’soriginal
definition for general Banach algebras ([12]) in the sense that it takes the
dual space structure of a von Neumann algebra into account. This notion of
amenabilitywaslaterdubbedConnes-amenabilitybyA.Ya.Helemski˘ı([11]).
In[18],theauthorextendedthenotionofConnes-amenabilitytothelarger
class of dual Banach algebras (a Banach algebra is called dual if it is a dual
Banachspacesuchthatmultiplicationisseparatelyw∗-continuous).Examples
ofdualBanachalgebras(besidesvonNeumannalgebras)are,forexample,the
measure algebras M(G) of locally compact groups G. In [20], the author
proved that a locally compact group G is amenable if and only if M(G) is
Connes-amenable–thusshowingthatthenotionofConnes-amenabilityisof
interestalsooutsidetheframeworkofvonNeumannalgebras.
In[8],E.G.EffrosshowedthatavonNeumannalgebraisConnes-amenable
ifandonlyifithasaso-callednormal,virtualdiagonal.LikeConnes-amena-
∗ResearchsupportedbyNSERCundergrantno.227043-00.
ReceivedJune3,2003;inrevisedformJuly10,2003.
dualbanachalgebras:connes-amenability,normal,virtual... 125
bility,thenotionofanormal,virtualdiagonaladaptsnaturallytothecontextof
generaldualBanachalgebras.ItisnothardtoseethatadualBanachalgebra
with a normal, virtual diagonal is Connes-amenable (the argument from the
vonNeumannalgebracasecarriesoveralmostverbatim;see[4]).
Let(cid:1)beadualBanachalgebrawith(notnecessarilyunique)predual(cid:1)∗;
itiseasytoseethat(cid:1)∗ isaclosedsubmoduleof(cid:1)∗.Considerthefollowing
threestatements:
(A) (cid:1)isConnes-amenable.
(B) (cid:1)hasanormal,virtualdiagonal.
(C) (cid:1)∗ isaninjective(cid:1)-bimoduleinthesenseof[10].
If (cid:1) is a von Neumann algebra, then (A), (B), and (C) are equivalent (the
equivalenceof(A)and(B)wasmentionedbefore;thattheyareequivalentto
(C) is proved in [11]). If (cid:1) = M(G) for a locally compact group G, then
(A) and (B) are also equivalent ([21]). For a general dual Banach algebra (cid:1),
weknowthat(B)implies(A),butnothingelseseemstobeknownaboutthe
relationsbetween(A),(B),and(C).
Asweshallseeinthepresentpaper,(C)alwaysimplies(B)–andthus(A)
– whereas the converse need not hold in general: this answers a question by
A.Ya. Helemski˘ı ([19, Problem 24]) in the negative. The counterexample is
themeasurealgebraM(G)foranyinfinite,amenable,locallycompactgroup
G; the proof relies on recent work by H. G. Dales and M. Polyakov ([6]).
(AsO.Yu.Aristovinformedusuponseeingapreprintversionofthispaper,it
hadpreviouslybeenshownbyS.TabaldyevthattheBanach(cid:8)1(G)-bimodule
c (G)isnotinjectiveforeveryinfinitediscretegroupG,whichalreadyanswers
0
Helemski˘ı’squestion;see[25].)
TheFourier-StieltjesalgebraB(G)ofalocallycompactgroupG,asintro-
duced in [9], is another example of a dual Banach algebra. In view of [20],
[21],and[22],itisnotfarfetchedtoconjecturethat(A)and(B)for(cid:1)=B(G)
areequivalentandholdtrueifandonlyifGhasanabeliansubgroupoffinite
index.Eventhoughwearenotabletosettlethisconjectureinfullgenerality,
wecancorroborateitforcertainG:(A)and(B)holdforB(G)–withGdis-
creteandamenableoratopologicalproductoffinitegroups–ifandonlyifG
hasanabeliansubgroupoffiniteindex.
Inthelastsectionofthepaperwemodifythedefinitionofanormal,virtual
diagonal by introducing what we call a σWC-virtual diagonal. For a dual
Banachalgebra(cid:1),wethenconsiderthestatement:
(B(cid:3)) (cid:1)hasaσWC-virtualdiagonal.
Unlikefor(A)and(B),wecanshowthat(A)and(B(cid:3))areindeedequivalent.
ItthusseemsthatthenotionofaσWC-virtualdiagonalseemstobethemore
126 volkerrunde
naturalonetoconsiderinthecontextofConnes-amenabilityifcomparedwith
thenotionofanormal,virtualdiagonal.
1. Preliminaries
1.1. Notions of amenability
WestartwiththedefinitionofadualBanachmodule:
Definition1.1. Let(cid:1)beaBanachalgebra.ABanach(cid:1)-bimoduleE is
calleddualifitisthedualofsomeBanachspaceE∗suchthat,foreacha ∈(cid:1),
themaps (cid:1)
a·x,
E →E, x (cid:6)→
x ·a
areσ(E,E∗)-continuous.
Remarks. 1.The predual space E∗ in Definition 1.1 need not be unique.
Nevertheless,E∗ willalwaysbeclearfromthecontext,sothatwecanspeak
ofthew∗-topologyonE withoutambiguity.
2.ItiseasilyseenthatadualBanachspaceE (withpredualE∗)whichis
alsoaBanach(cid:1)-bimoduleisadualBanach(cid:1)-bimoduleifandonlyifE∗ isa
closedsubmoduleofE∗.Hence,ourdefinitionofadualBanach(cid:1)-bimodule
coincideswiththeusualone(givenin[19],forinstance).
Let(cid:1)beaBanachalgebra,andletEbeaBanach(cid:1)-bimodule.Aderivation
from(cid:1)toE isabounded,linearmapD:(cid:1)→E satisfying
D(ab)=a·Db+(Da)·b (a,b ∈(cid:1)).
AderivationD:(cid:1)→E iscalledinner ifthereisx ∈E suchthat
Da =a·x −x ·a (x ∈(cid:1)).
Definition1.2. ABanachalgebra(cid:1)iscalledamenableifeveryderivation
from(cid:1)intoadualBanach(cid:1)-bimoduleisinner.
The terminology is, of course, motivated by [12, Theorem 2.5]:A locally
compactgroupGisamenableifandonlyifitsgroupalgebraL1(G)isamenable
inthesenseofDefinition1.2.
ForsomeclassesofBanachalgebra,Definition1.2seemstobe“toostrong”
in the sense that it only characterizes fairly uninteresting examples in those
classes:AvonNeumannalgebraisamenableifandonlyifitissubhomogen-
eous ([26]), and the measure algebra M(G) of a locally compact group G is
amenableifandonlyifGisdiscreteandamenable([5]).
BothvonNeumannalgebrasandmeasurealgebrasaredualBanachalgebras
inthesenseofthefollowingdefinition:
dualbanachalgebras:connes-amenability,normal,virtual... 127
Definition1.3. ABanachalgebra(cid:1)whichisadualBanach(cid:1)-bimodule
iscalledadualBanachalgebra.
Examples. 1.EveryvonNeumannalgebraisadualBanachalgebra.
2.ThemeasurealgebraM(G)ofalocallycompactgroupGisadualBanach
algebra(withpredualC (G)).
0
3. If E is a reflexive Banach space, then B(E) is a dual Banach algebra
(withpredualE⊗ˆ E∗,where⊗ˆ denotestheprojectivetensorproductofBanach
spaces).
4. The bidual of every Arens regular Banach algebra is a dual Banach
algebra.
WeshallnowintroduceavariantofDefinition1.2fordualBanachalgebras
thattakesthedualspacestructureintoaccount:
Definition1.4. Let(cid:1)beadualBanachalgebra,andletEbeadualBanach
(cid:1)-bimodule.Anelementx ∈E iscallednormalifthemaps
(cid:1)
a·x,
(cid:1)→E, a (cid:6)→
x ·a
arew∗-continuous.ThesetofallnormalelementsinE isdenotedbyEσ.We
saythatE isnormalifE =Eσ.
Remark. Itiseasytoseethat,foranydualBanach(cid:1)-bimoduleE,theset
Eσ isanormclosedsubmoduleofE.Generally,however,thereisnoneedfor
Eσ tobew∗-closed.
Definition 1.5. A dual Banach algebra (cid:1) is called Connes-amenable if
everyw∗-continuousderivationfrom(cid:1)intoanormal,dualBanach(cid:1)-bimodule
isinner.
Remarks. 1. “Connes”-amenability was introduced by B. E. Johnson, R.
V. Kadison, and J. Ringrose for von Neumann algebras in [15]. The name
“Connes-amenability” seems to originate in [11], probably in reverence to-
wardsA.Connes’fundamentalpaper[2].
2.ForavonNeumannalgebra,Connes-amenabilityisequivalenttoanum-
berofimportantproperties,suchasinjectivityandsemidiscreteness;see[19,
Chapter6]forarelativelyself-containedaccount.
3. The measure algebra M(G) of a locally compact group G is Connes-
amenableifandonlyifGisamenable([20]).
128 volkerrunde
1.2. Virtual diagonals
Let(cid:1)beaBanachalgebra.Then(cid:1)⊗ˆ (cid:1)isaBanach(cid:1)-bimodulevia
a·(x ⊗y):=ax ⊗y and (x ⊗y)·a :=x ⊗ya (a,x,y ∈(cid:1)),
sothatthemultiplicationmap
(cid:19):(cid:1)⊗ˆ (cid:1)→(cid:1), a⊗b (cid:6)→ab
becomesahomomorphismofBanach(cid:1)-bimodules.
ThefollowingdefinitionisalsoduetoB.E.Johnson([13]):
Definition1.6. AvirtualdiagonalforaBanachalgebra(cid:1)isanelement
M ∈((cid:1)⊗ˆ (cid:1))∗∗ suchthat
a·M =M·a and a(cid:19)∗∗M =a (a ∈(cid:1)).
In[13], JohnsonshowedthataBanachalgebra(cid:1)isamenableifandonly
if it has a virtual diagonal. This allows to introduce a quantified notion of
amenability:
Definition1.7. ABanachalgebra(cid:1)iscalledC-amenableforsomeC ≥1
ifithasavirtualdiagonalofnormatmostC.TheinfimumoverallC ≥1such
that(cid:1)isC-amenableiscalledtheamenabilityconstant of(cid:1)anddenotedby
AM .
(cid:1)
Remark. It follows from the Alaoglu–Bourbaki theorem ([7, Theorem
V.4.2]),thattheinfimuminthedefinitionofAM isattained,i.e.isaminimum.
(cid:1)
Definition 1.6 has a variant that is better suited for dual Banach algebras.
Let(cid:1)beadualBanachalgebrawithpredual(cid:1)∗,andletBσ2((cid:1),C)denotethe
bounded,bilinearfunctionalson(cid:1)×(cid:1)whichareseparatelyw∗-continuous.
Since(cid:19)∗ maps(cid:1)∗ intoBσ2((cid:1),C),itfollowsthat(cid:19)∗∗ dropstoan(cid:1)-bimodule
homomorphism(cid:19)σ:Bσ2((cid:1),C)∗ →(cid:1).Wedefine:
Definition1.8. Anormal,virtualdiagonal foradualBanachalgebra(cid:1)
isanelementM ∈B2((cid:1),C)∗ suchthat
σ
a·M =M·a and a(cid:19)σM =a (a ∈(cid:1)).
Remarks. 1.EverydualBanachalgebrawithanormal,virtualdiagonalis
Connes-amenable([4]).
2.AvonNeumannalgebraisConnes-amenableifandonlyifithasanormal,
virtualdiagonal([8]).
dualbanachalgebras:connes-amenability,normal,virtual... 129
3. The same is true for the measure algebras of locally compact groups
([21]).
In[18],weintroducedastrongervariantofDefinition1.5–called“strong
Connes-amenability” – and showed that the existence of a normal, virtual
diagonalforadualBanachalgebrawasequivalenttoitbeingstronglyConnes-
amenable([18,Theorem4.7]).Thefollowingproposition,observedbythelate
B.E.Johnson,showsthatstrongConnes-amenabilityisevenstrongerthanit
seems:
Proposition1.9.ThefollowingareequivalentforadualBanachalgebra
(cid:1):
(i) Thereisanormal,virtualdiagonalfor(cid:1).
(ii) (cid:1) has an identity, and every w∗-continuous derivation from (cid:1) into a
dual,unitalBanach(cid:1)-bimoduleisinner.
Proof. Inviewof[18,Theorem4.7],only(i)(cid:14)⇒(ii)needsproof.
LetEbeadual,unitalBanach(cid:1)-bimodule.Dueto[18,Theorem4.7],itis
sufficienttoshowthatD(cid:1)⊂Eσ.This,however,isautomaticallytruebecause
a·Db =D(ab)−(Da)·b and (Db)·a =D(ab)−b·Da (a,b ∈(cid:1))
holds.
1.3. Injectivity for Banach modules
Let (cid:1) be a Banach algebra, and let E be a Banach space. Then B((cid:1),E)
becomesaleftBanach(cid:1)-bimodulebyletting
(a·T)(x):=T(xa) (a,x ∈(cid:1)).
IfEisalsoaleftBanach(cid:1)-module,thereisacanonicalmodulehomomorph-
ismι:E →B((cid:1),E),namely
ι(x)a :=a·x (x ∈E, a ∈(cid:1)).
Forthedefinitionofinjective,leftBanachmodulesdenote,foranyBanach
algebra(cid:1),by(cid:1)# theunconditionalunitization,i.e.weadjoinanidentityto(cid:1)
nomatterif(cid:1)alreadyhasoneornot.Clearly,ifEisaleftBanach(cid:1)-module,
themoduleoperationextendscanonicallyto(cid:1)#.
Definition1.10. Let(cid:1)beaBanachalgebra.AleftBanach(cid:1)-moduleE
iscalledinjectiveifι:E →B((cid:1)#,E)hasaboundedleftinversewhichisalso
aleft(cid:1)-modulehomomorphism.
130 volkerrunde
Therearevariousequivalentconditionscharacterizinginjectivity(see,e.g.,
[19,Proposition5.3.5]).Thefollowingis[6,Proposition1.7]:
Lemma1.11. Let(cid:1)beaBanachalgebra,andletEbeafaithfulleftBanach
(cid:1)-module,i.e.ifx ∈ E issuchthata·x = 0foralla ∈ (cid:1),thenx = 0.Then
E isinjectiveifandonlyifι:E →B((cid:1),E)hasaboundedleftinversewhich
isalsoan(cid:1)-modulehomomorphism.
Definition 1.10 and Lemma 1.11 can be adapted to the context of right
modulesandbimodulesinastraightforwardway.
TherelevanceofinjectivityinthecontextofamenableBanachalgebrasbe-
comesapparentfrom[10,TheoremVII.2.20]andthedualitybetweeninjectiv-
ityandflatness([10,TheoremVII.1.14]):ABanachalgebra(cid:1)withbounded
approximateidentityisamenableifandonlyiftheBanach(cid:1)-bimodule(cid:1)∗ is
injective.
2. Injectivity of the predual bimodule
InviewofthecharacterizationofamenableBanachalgebrasjustmentioned,
one might ask if an analogous statement holds for Connes-amenable, dual
Banachalgebras(cid:1)with(cid:1)∗ replacedby(cid:1)∗.ForvonNeumannalgebras, this
isknowntobetrue([11]).
OurfirstresultistrueforalldualBanachalgebras:
Proposition2.1. Let(cid:1)beadualBanachalgebrawithidentitysuchthat
itspredualbimodule(cid:1)∗ isinjective.Then(cid:1)hasanormal,virtualdiagonal.
Proof. Considertheshortexactsequence
(cid:19)∗|
(1) {0}→(cid:1)∗ −→(cid:1)∗ Bσ2((cid:1),C)→Bσ2((cid:1),C)/(cid:19)∗(cid:1)∗ →{0}.
DefineP:Bσ2((cid:1),C)→(cid:1)∗ byletting
(P(cid:25))(a):=(cid:25)(a,e ) ((cid:25)∈B2((cid:1),C), a ∈(cid:1)),
(cid:1) σ
where e denotes the identity of (cid:1). Then it is routinely checked that P is a
(cid:1)
bounded projection onto (cid:19)∗(cid:1)∗ and thus a left inverse of (cid:19)∗|(cid:1)∗. Hence, (1)
is admissible ([19, Definition 2.3.12]). Since (cid:1)∗ is an injective (cid:1)-bimodule,
thereisabounded(cid:1)-bimodulehomomorphismρ:Bσ2((cid:1),C) → (cid:1)∗ whichis
a left inverse of (cid:19)∗| ([19, Proposition 5.3.5]). It is routinely checked that
(cid:1)∗
ρ∗(e )isanormal,virtualdiagonalfor(cid:1).
(cid:1)
As we shall soon see, the converse of Proposition 2.1 is, in general, false.
Nevertheless,forcertain(cid:1),theinjectivityof(cid:1)∗isindeedequivalenttotheex-
istenceofanormalvirtualdiagonalfor(cid:1)(andeventoitsConnes-amenability).
dualbanachalgebras:connes-amenability,normal,virtual... 131
Wefirstrequirealemma:
Lemma2.2. Let(cid:1)beaBanachalgebrawithidentity,letI beaclosedideal
of(cid:1),andletE beaunitalBanach(cid:1)-bimodulesuchthat:
(a) E isinjectiveasaBanachI-bimodule.
(b) E isfaithfulbothasaleftandarightBanachI-module.
ThenE isinjectiveasaBanach(cid:1)-bimodule.
Proof. TurnB((cid:1)⊗ˆ (cid:1),E)intoaBanach(cid:1)-bimodule,byletting
(a·T)(x⊗y):=T(x⊗ya) and (T·a)(x⊗y):=T(ax⊗y) (a,x,y ∈(cid:1)).
Defineι:E →B((cid:1)⊗ˆ (cid:1),E)byletting
ι(x)(a⊗b):=a·x ·b (x ∈E, a,b ∈(cid:1)).
Since(cid:1)hasanidentityandEisunital,itissufficientby(thebimoduleanalogue
of) Lemma 1.11 to show that ι has a bounded left inverse which is an (cid:1)-
bimodulehomomorphism.
By(a),ιhasaboundedleftinverseρ whichisanI-bimodulehomomorph-
ism.We claim that ρ is already an (cid:1)-bimodule homomorphism. To see this,
leta ∈ (cid:1),T ∈ B((cid:1)⊗ˆ (cid:1),E),andb ∈ I.SinceI isanidealof(cid:1),weobtain
that
b·ρ(a·T)=ρ(ba·T)=ba·ρ(T),
sothatb·(ρ(a·T)−a·ρ(T))=0.Sinceb ∈I wasarbitrary,andsinceEis
afaithfulleftBanachI-moduleby(b),weobtainρ(a·T) = a·ρ(T);since
a ∈ (cid:1) and T ∈ B((cid:1)⊗ˆ (cid:1),E) were arbitrary, ρ is therefore a left (cid:1)-module
homomorphism.
Analogously,oneshowsthatρ isaright(cid:1)-modulehomomorphism.
Ourfirsttheorem,considerablyimproves[18,Theorem4.4]:
Theorem2.3. Let(cid:1)beanArensregularBanachalgebrawhichisanideal
in(cid:1)∗∗.Thenthefollowingareequivalent:
(i) (cid:1)isamenable.
(ii) (cid:1)∗ isaninjectiveBanach(cid:1)∗∗-bimodule.
(iii) (cid:1)∗∗ hasanormal,virtualdiagonal.
(iv) (cid:1)∗∗ isConnes-amenable.
Proof. (i) (cid:14)⇒ (ii): We wish to apply Lemma 2.2. Since (cid:1) is amenable,
it has a bounded approximate identity. TheArens regularity of (cid:1) yields that
(cid:1)∗∗ has an identity and that (cid:1)∗ is a unital Banach (cid:1)∗∗-bimodule. Since (cid:1) is
132 volkerrunde
amenableandthusaflat(cid:1)-bimoduleoveritself([10,TheoremVII.2.20]),(cid:1)∗
isaninjectiveBanach(cid:1)-bimoduleby(thebimoduleversionof)[10,Theorem
VII.1.14].Thus,Lemma2.2(a)issatisfied.ToseethatLemma2.2(b)holdsas
well, let φ ∈ (cid:1)∗ \{0}. Choose a ∈ (cid:1) such that (cid:21)a,φ(cid:22) (cid:23)= 0. Let (eα)α be a
boundedapproximateidentityfor(cid:1).Sincelimα(cid:21)aeα,φ(cid:22) = (cid:21)a,φ(cid:22) (cid:23)= 0,there
isb ∈(cid:1)suchthat(cid:21)ab,φ(cid:22)=(cid:21)a,b·φ(cid:22)(cid:23)=0andthusb·φ (cid:23)=0.Consequently,(cid:1)∗
isfaithfulasaleftBanach(cid:1)-module.Analogously,oneverifiesthefaithfulness
of(cid:1)∗ asarightBanach(cid:1)-module.
(ii)(cid:14)⇒(iii)isclearbyProposition2.1.
(iii)(cid:14)⇒(iv)holdsby[4].
(iv)(cid:14)⇒(i):Thisisonedirectionof[18,Theorem4.4].
Example. LetEbeareflexiveBanachspacewiththeapproximationprop-
erty,andlet(cid:1)beK(E),thealgebraofallcompactoperatorsonE.Then(cid:1)∗
canbecanonicallyidentifiedwithN(E∗),thenuclearoperatorsonE∗,andwe
have(cid:1)∗∗ =B(E).ByTheorem2.3,wehavetheequivalenceofthefollowing
properties:
(i) K(E)isamenable.
(ii) N(E∗), the space of nuclear operators on E∗, is an injective Banach
B(E)-bimodule.
(iii) B(E)hasanormal,virtualdiagonal.
(iv) B(E)isConnes-amenable.
InviewofthesituationforvonNeumannalgebras,onemightbetemptedby
Theorem2.3tojumptotheconclusionthat,foradualBanachalgebra(cid:1)with
predual(cid:1)∗,theinjectivityof(cid:1)∗isequivalentto(cid:1)beingConnes-amenableor
havinganormal,virtualdiagonal.
Our next theorem reveals that this is not the case: this gives a negative
answertoaquestionposedbyA.Ya.Helemski˘ı([19,Problem24]).
Lemma2.4. LetGbealocallycompactgroup,andsupposethatC (G)is
0
injectiveasaleftBanachM(G)-module.ThenC (G)isalsoinjectiveasaleft
0
BanachL1(G)-module.
Proof. For (cid:2) = M(G) or (cid:2) = L1(G), turn B((cid:2),C (G)) into a left
0
BanachM(G)-moduleinthecanonicalway,andletι :C (G)→B((cid:2),C (G))
(cid:2) 0 0
betherespectivecanonicalleftM(G)-modulehomomorphism.
Since C (G) is a unital left M(G)-module, it is immediate that the ho-
0
momorphism ιM(G):C0(G) → B(M(G),C0(G)) of left modules, has a lin-
ear, bounded left inverse. Since C (G) is injective as a left Banach M(G)-
0
module,ιM(G) hasaleftinverseρ whichisaboundedhomomorphismofleft
M(G)-modules. Let T ∈ B(M(G),C0(G)) be such that T|L1(G) = 0. Since
dualbanachalgebras:connes-amenability,normal,virtual... 133
L1(G) is an ideal in M(G), it follows from the definition of the module ac-
tion on B(M(G),C (G)), that f ·T = 0 for all f ∈ L1(G) and therefore
0
f ·ρ(T) = ρ(f ·T) = 0 for all f ∈ L1(G). Since C (G) is a faithful left
0
L1(G)-module, this means that ρ(T) = 0. Since L1(G) is complemented in
M(G), it follows that ρ:B(M(G),C (G)) → C (G) drops to bounded ho-
0 0
momorphism of left M(G)-modules ρ˜:B(L1(G),C (G)) → C (G), which
0 0
iseasilyseentobealeftinverseofιL1(G).
Sinceρ˜ istriviallyahomomorphismofleftL1(G)-modules,Lemma1.11
yieldstheinjectivityofC (G)asaleftBanachL1(G)-module.
0
Theorem2.5. LetGbealocallycompactgroup.ThenC (G)isaninjective
0
BanachM(G)-bimoduleifandonlyifGisfinite.
Proof. Suppose that C (G) is an injective Banach M(G)-bimodule. By
0
[10, Proposition VII.2.1] and Lemma 2.4, C (G) is also injective as a left
0
BanachL1(G)-module.By[6,Theorem3.8],thismeansthatGmustbefinite.
Theconverseisobvious.
Remark. In contrast, it was proven in [20], for a locally compact group
G, that M(G) is Connes-amenable – and, equivalently, has a normal, virtual
diagonalby[21]–ifandonlyifGisamenable.
3. Fourier-Stieltjes algebras of locally compact groups
The Fourier-Stieltjes algebra B(G) of a locally compact group G was intro-
duced by P. Eymard in [9] along with the Fourier algebra A(G).We refer to
[9] for further information on these algebras. It is straightforward to see that
B(G)isadualBanachalgebra–withpredualC∗(G)–foranylocallycompact
groupGwhereasA(G)neednotevenbeadualspace(unlessGiscompact,
ofcourse).
Let G be a locally compact group G with an abelian subgroup of finite
index.ThenA(G)isamenableandw∗-denseinB(G),sothatB(G)isConnes-
amenable.Infact,aformallystrongerconclusionholds:
Proposition3.1. LetGbealocallycompactgroupwithanabeliansub-
groupoffiniteindex.ThenB(G)hasanormal,virtualdiagonal.
Proof. LetH beaanabeliansubgroupofGsuchthatn:=[G:H]<∞.
ReplacingH byitsclosure,wemaysupposethatH isclosedandthusopen.
Consequently,therestrictionmapfromB(G)ontoB(H)issurjectivesothat
B(G)∼=B(H)n ∼=M(Hˆ)n,
whereHˆ isthedualgroupofH.By[21],M(Hˆ)hasanormal,virtualdiagonal.
It is easy to see that therefore M(Hˆ)n ∼= B(G) must have a normal, virtual
diagonalaswell.