A simple characterisation of topological amenability in terms of bounded coho mology is proved following Johnsons formulation of amenability The connection to injective Banach modules is established 1 Introduction 1A Motivation According to JohnsonR ID: 55740
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2NICOLASMONODamenability.Thisre ectsthefactthatnotall(G;X)-modulesaremodulesfortheunderlyingcrossedproductalgebraintroducedbelow.AnalysingonlytheC(X)-structurealone,wecanhoweverisolatearelevantconceptinspiredbyKakutani'sclassicalwork[19,20]:Denition1.WesaythataC(X)-moduleEisoftypeMifforallu2Eand'i2C(X)with'i0onehas(M)k'1uk++k'nukk'1++'nkkuk:WesaythatEisoftypeCifforallui2Eand'i0onehas(C)k'1u1++'nunkk'1++'nkmaxikuik:(Thesenotionsaremutuallydual,yetbothtypesincludemodulesthatarenotdual.Thefunda-mentalexamplesmotivatingtheterminologyaremodulesofmeasures,respectivelyofcontinuousfunctions.)Wenowhavethecriterionthatwesought:Proposition(Conciseversion).LetGbeagroupactingonthecompactspaceX.Thefollowingareequivalent.(i)TheG-actiononXistopologicallyamenable.(ii)H1b(G;E)=0forevery(G;X)-moduleEoftypeM.(iii)Hnb(G;E)=0forevery(G;X)-moduleEoftypeMandeveryn1.1.C.Injectivemodulesandmeans.OurcohomologicalcharacterisationextendstoafurtheraspectnotcoveredbyJohnson{Ringrose,namelyinjectivemodules(seex2.A).Bywayofmotiva-tion,werecalltheanalogoussituationinergodictheory:Formeasurespaces,amenableactionsinZimmer'ssense[27]arecharacterisedbytherelativeinjectivityofthecorrespondingmodulesofL1-maps[6,23].Weshallprovideasimilarstate-mentfortopologicalamenability.However,thenotionof\boundedmeasurablemaps"becomesproblematicwhenXisnon-metrisable,asitwilldenitelybeforapplicationstoexactnesswhereXistheCech{StonespaceG.Wesubmitthatagoodanalogueisprovidedincompletegener-alitybythedualofthespaceI(C(X);W)ofintegraloperatorsvaluedinanyBanachspaceW(inthesenseofGrothendieck[13,x2]forabstractintegraloperatorsbetweenarbitraryBanachspaces).Hencethestatements(iii)and(vii)below.Asitturnsout,thiscriterionappliestoalldualmodulesoftypeC,see(viii).Finally,asintheJohnson{Ringrosecriterion,wetransitthroughacharacterisationofamenabil-ityintermsofinvariantmeans.Theequivalenceofthiswiththedenitionofamenabilityisroutine,butwederiveitforcompleteness,see(ix)and(x)below.ThesummationconditionthereinmeansthatthecanonicalimageinC(X)oftheinvariantelementisthebidualoftheconstantfunction1X2C(X).Proposition(Fullversion).LetGbeagroupactingonthecompactspaceX.Thefollowingareequivalent.(i)TheG-actiononXistopologicallyamenable.(ii)Hnb(G;C(X;V))=0foreveryBanachG-moduleVandeveryn1.(iii)Hnb(G;I(C(X);W))=0foreveryBanachG-moduleWandeveryn1.(iv)Hnb(G;E)=0forevery(G;X)-moduleEoftypeMandeveryn1.(v)Anyofthepreviousthreepointsholdsforn=1.(vi)C(X;V)isrelativelyinjectiveforeveryBanachG-moduleV.(vii)I(C(X);W)isrelativelyinjectiveforeveryBanachG-moduleW.(viii)Everydual(G;X)-moduleoftypeCisarelativelyinjectiveBanachG-module.(ix)ThereisaG-invariantelementinC(X;`1G)summingto1X.(x)ThereisanormonepositiveG-invariantelementinC(X;`1G)summingto1X.Remarks.(a)Theproofshowsthatitsucesin(ix)and(x)thatthesummationbeanyfunctionthatnevervanishes.Thenormconditionin(x)isredundant. 4NICOLASMONODA-normofsuchanelementis Xg'g g A= Xgj'gj C(X)The(G;X)-structureturnsAintoaBanachalgebrathatwecalltheBanachcrossedproductofGandX,nottobeconfusedwiththeC*-algebraiccrossedproduct.Explicitly,theproductis(' g)( h)='g ghonelementarytensorswithg;h2Gand'; 2C(X).Thenaturalalgebraicinvolution' g7!g�1' g�1doesnotingeneralextendtoA(Example9),andthereisindeedafundamentaldi erencebetweenrightandleftA-modules,asweshallsee.Proposition5.LetGbeagroupactingonthecompactspaceXandletEbea(G;X)-module.LetA=C(X;`1G).(i)IfEisoftypeC,thenitisnaturallyaleftA-module.(ii)IfEisoftypeM,thenitisnaturallyarightA-module.Example6.The(G;X)-moduleC(X)^ `1G,where^ istheprojectivetensorproduct,isnotanA-moduleunlesseitherGorXisnite.ThisisaspecialcaseoftheDvoretzky{RogersTheorem[10]sinceC(X)^ `1G=`1(G;C(X)).InparticularitisnotoftypeC;inaddition,itisnotoftypeMeitherunlessXisapoint.ProofofProposition5.Thecondition(C)preciselyshowsthatfornitesums =Pg g gandu2Ewehavek ukE= Xg g(gu) Xgj gj C(X)maxgkgukE=k kAkukEandthustheleftmultiplicationextendstoA.IfEisoftypeM,wedeneu :=Xg�(g�1 g) g�1u=Xgg�1( gu);whichisarightmodulestructureovernitesums.Wethencarryoutasimilarcomputationusingtheinequality(M).3.AmenableactionsTheactionofagroupGonacompactspaceXiscalledtopologicallyamenableifthereisanetfjgj2JinC(X;`1G)suchthateveryj(x)isaprobabilitymeasureonGandlimj2J gj�j C(X;`1G)=0(8g2G):NoticethatifGiscountable,thenetcanbereplacedbyasequence;thisdoesnotrequireXtobemetrisableandhenceappliesforinstancetotheCech{Stonecompacticationofacountablegroup.Noticealsothatthecontinuityofj:X!`1Gcanbereplacedbyweak-*continuitysincebothtopologiescoincideonprobabilitymeasures.BycompactnessofX,onecanassumethatthesupportofj(x)isinanitesetdependingonlyonj.Finally,weobservethat,asexpected,amenabilityisan\approximateproperness"withjbeinganapproximateBruhatfunction.Backgroundreferencesare[1,4];werecallthattheamenabilityoftheactionisequivalenttothenuclearityoftheC*-algebraicreducedcrossedproduct,andthusinturntoitsamenability,seeConnes[7]andHaagerup[15].3.A.Proofs.Inordertotakeadvantageofsomesimpleimplications,theproofwillnotfollowasinglecycleofimplications.Theheartofthematteristhefollowingimplication,whichstillholdsforGlocallycompactsecondcountablewiththedenitionsof[1,x2]and[23,x4.1]. 6NICOLASMONOD(i))(x)isobtainedbychoosingaweak-*accumulationpointandkeepinginmindthepropertiesofBanachlatticesrecalledabove.Since(x))(ix)istautological,thiscompletestheproof.4.ExactgroupsandfurtherremarksThedenitionofexactgroupsgoesbacktoKirchberg{Wassermann[21];itisequivalenttothestatementthatthereducedC*-algebraofthegroupisexactforspatialtensorproducts.Ozawa[26]andAnantharaman-Delaroche[1]provedthatthegroupGisexactifandonlyifitsactionontheCech{StonecompacticationGisamenable.Higson{Roe[16]establishedthatthelatterconditionisequivalenttoYu'sPropertyA.Exactnesshasdeepimplicationsforthegroupbutatthesametimeitisnotoriouslydiculttoproduceanynon-exactgroup;Gromovsucceededin[12]usingtherelationtouniformembeddingsintoHilbertspaces.InviewoftheequivalencewithamenabilityoftheactiononG,thepropositionsoftheintro-ductioncharacteriseexactgroups.Inthisspecialcase,C(G)=`1G,allowingsomeadditionalidentications.Forinstance,(G;G)-modulesareparticularlyconcreteandcanbethoughtofasmodulesovertheinvolutiveBanach{Hopfalgebra`1G.Example7.LetVbeaBanachG-module.AnyG-invariant`1G-submoduleof`1(G;V)isa(G;G)-moduleoftypeC.ItisgenerallynotoftheformC(G;V).ThisexampleprovidesalinktotheworkofDouglas{Nowak[9].Anotherarticleinvestigatingcohomologicalcharacterisationsofexactnessis[3].Anotherobviousexampleof(G;G)-moduleisprovidedbythevarious`pG.Example8.LetGbeanon-Abelianfreegroup.ThentheG-actiononX:=Gisamenable.Ontheotherhand,H2b(G;`1G)isnon-trivial,eventhough`1Gisadual(G;X)-module.Infact,itisoftypeMandisthedualofthe(G;X)-modulec0(G)oftypeC.Moregenerally,H2b(G;`pG)isnon-trivialforallp1usingtheargumentof[25].ThealgebraA:=C(G;`1G)=`1G `1G=ucG[`1G];whichcoincideswiththe\uniformalgebra"`uGof[9],alsoadmitsadditionalrealizationssincenowanyelementcanbeviewedasakernelonGG.Wendthus:A=C(`1G)=Lw*/w(`1G);whereCdenotescompactoperatorsandLw*/wtheweak-*-to-weakcontinuousoperators(whicharenecessarilycompact),comparep.90{91inGrothendieck[14].(Theproduct,meanwhile,remainsG-twisted).Example9.TheinvolutionfamiliarfromC*-crossedproductsisnotA-bounded:ifSGisaniteset,thediagonalPg2Sg ghasunitnormbutitsimagee Pg2Sg�1hasnormjSj.ReturningtoG-actionsongeneralcompactspacesX,weconcludewithsuggestionsforfurtherinvestigation.Question10.LetEbea(G;X)-module.InthespiritofKakutani[19,20],giveaconcreteexpressionoftheintrinsicpropertiesCandM.Questions11.(a)CharacterisetheamenabilityofA:=C(X;`1G).(b)Characterise(anddene!)theco-amenabilityofthesubalgebra`1GinA:=C(X;`1G),embeddedbytensoringwith1X.Anaturalcandidateforthedenitionofco-amenablesubalgebrasis:TheinclusionmorphisminducesaninjectiverestrictionmorphismatthelevelofBanachalgebracohomology,foralldualmodulesoverthetarget.(ThisisthefaithfultranslationofProp.3in[24].)