The Rokhlin property for automorphisms on simple C

arXiv:math/0602513v1[math.OA]23Feb2006TheRokhlinPropertyforAutomorphismsonSimpleC∗-algebrasHuaxinLinAbstractWestudyageneralKishimoto’sproblemforautomorphismsonsimpleC∗-algebraswithtracialrankzero.LetAbeaunitalseparablesimpleC∗-algebrawithtracialrankzeroandletαbeanautomorphism.UndertheassumptionthatαhascertainRokhlinproperty,wepresentaproofthatA⋊αZhastracialrankzero.Wealsoshowthatiftheinducedmapα∗0onK0(A)fixesa“dense”subgroupofK0(A)thenthetracialRokhlinpropertyimpliesastrongerRokhlinproperty.Consequently,theinducedcrossedproductC∗-algebrashavetracialrankzero.1IntroductionTheRokhlinpropertyinergodictheorywasadoptedtothecontextofvonNeumannalgebrasbyConnes([3]).TheRokhlinproperty(withvariousversions)wasalsointroducedtothestudyofautomorphismsonC∗-algebras(see,forexample,HermanandOcneanu([13]),Rørdam([28],Kishimoto([10])andPhillips[27]amongothers–seealsothenextsection).AconjectureofKishimotocanbeformulatedasfollows:LetAbeaunitalsimpleAT-algebraofrealrankzeroandαbeanapproximatelyinnerautomorphism.Supposethatαis“sufficientlyouter”,thenthecrossedproductoftheAT-algebrabyα,A⋊αZ,isagainaunitalAT-algebra.Kishimotoshowedthatthisistrueforanumberofcases,inparticular,somecasesthatAhasauniquetracialstate.KishimotoproposedthattheappropriatenotionofouternessistheRokhlinproperty([11]).Kishimoto’sproblemhasamoregeneralsetting:P1LetAbeaunitalseparablesimpleC∗-algebrawithtracialrankzeroandletαbeanautomorphism.SupposethatαhasaRokhlinproperty.DoesA⋊αZhavetracialrankzero?IfinadditionAisassumedtobeamenableandsatisfytheUniversalCoefficientTheorem,then,bytheclassificationtheorem([20]),AisanAH-algebrawithslowdimensiongrowthandwithrealrankzero.IfA⋊αZhastracialrankzero,then,again,by[20],A⋊αZisanAH-algebra(withslowdimensiongrowthandwithrealrankzero).NotethatsimpleAT-algebraswithrealrankzeroareexactlythosesimpleAH-algebraswithtorsionfreeK-theory,withslowdimensiongrowthandwithrealrankzero.Itshouldalsobenoted([18])thataunitalsimpleAH-algebrahasslowdimensiongrowthandrealrankzeroifandonlyifithastracialrankzero.IfKi(A)aretorsionfreeandαisapproximatelyinner,thenKi(A⋊αZ)istorsionfree.ThusanaffirmativeanswertotheproblemP1provestheoriginalKishimoto’sconjecture(seealso[23]).Oneshouldalsonoticethatifαisnotapproximatelyinner,Ki(A⋊αZ)mayhavetorsionevenifAdoesnot.Therefore,inKishimoto’sproblem,therestrictionthatαisapproximatelyinnercannotberemoved.SoitisappropriatetoreplaceAT-algebrasbyAH-algebrasiftherequirementthatαisapproximatelyinnerisremoved.Inthispaperwereportsomeoftherecentdevelopmentonthissubject.LetAbeaunitalseparablesimpleC∗-algebrawithtracialrankzeroandletαbeanau-tomorphismonA.Insection3,wepresentaproofthatifαsatisfiesthetracialcyclicRokhlinproperty(see2.4below)thenthecrossedproductA⋊αZhastracialrankzerowhichgivesa1solutiontoP1,providedthatαr∗0|G=idG(forsomeintegerr≥1)forsome“dense”subgroupofK0(A).Insection4,wediscusswhenαhasthetracialcyclicRokhlinproperty.ThetracialRokhlinpropertyintroducedbyN.C.Phillips(see[27]and[26])hasbeenprovedtobeanatu-ralgeneralizationoforiginalRokhlintowersforergodicactions.Weprovethat,ifinaddition,αr∗0|G=id|G(forsomeintegerr≥1)forsomesubgroupG⊂K0(A)forwhichρA(G)isdenseinρA(K0(A)),thenthatαhastracialRokhlinpropertyimpliesthatαhastracialcyclicRokhlinproperty.ByTheorem3.4insection2,theassociatedcrossedproductA⋊αZhastracialrankzero.Itshouldbepointedoutthat,intheoriginalKishimoto’sconjecture,αisassumedtobeapproximatelyinner.Inthatcase,α∗0=idK0(A).Wealsoshowthat,theconditionthatα∗0|G=idGforsome“dense”subgroupGissometimeautomatic.AcknowledgementThispaperwasintendedfortheProceedingsofGPOTS2005.MostofitwaswrittenwhentheauthorwasinEastChinaNormalUniversityduringthesummer2005.ItispartiallysupportedbyagrantfromNSFofU.S.A,ShanghaiPriorityAcademicDisciplinesandZhi-JiangProfessorshipfromEastChinaNormalUniversity.2PreliminariesWewillusethefollowingconvention:(i)LetAbeaC∗-algebra,leta∈Abeapositiveelementandletp∈Abeaprojection.Wewrite[p]≤[a]ifthereisaprojectionq∈aAaandapartialisometryv∈Asuchthatv∗v=pandvv∗=q.(ii)LetAbeaC∗-algebra.WedenotebyAut(A)theautomorphismgroupofA.IfAisunitalandu∈Aisaunitary,wedenotebyadutheinnerautomorphismdefinedbyadu(a)=u∗auforalla∈A.(iii)LetT(A)bethetracialstatespaceofaunitalC∗-algebraA.Itisacompactconvexset.DenotebyAff(T(A))thenormedspaceofallrealaffinecontinuousfunctionsonT(A).DenotebyρA:K0(A)→Aff(T(A))thehomomorphisminducedbyρA([p])(τ)=τ(p)forτ∈T(A).Itshouldbenoted,by[2],ifAisaunitalsimpleamenableC∗-algebrawithrealrankzeroandstablerankandweaklyunperforatedK0(A),ρA(K0(A))isdenseinAff(T(A)).(iv)LetAandBbetwoC∗-algebrasandϕ,ψ:A→Bbetwomaps.Letǫ0andF⊂Abeafinitesubset.Wewriteϕ≈ǫψonF,ifkϕ(a)−ψ(a)kǫforalla∈F.(v)Letx∈A,ǫ0andF⊂A.Wewritex∈ǫF,ifdist(x,F)ǫ,orthereisy∈Fsuchthatkx−ykǫ.(vi)Ifh:A→Bisahomomorphism,thenh∗i:Ki(A)→Ki(B)(i=0,1)istheinducedhomomorphism.WerecallthedefinitionoftracialtopologicalrankofC*-algebras.Definition2.1.[15,Theorem6.13]LetAbeaunitalsimpleC∗-algebra.ThenAissaidtohavetracial(topological)rankzeroifforanyfinitesetF⊂A,andǫ0andanynon-zeropositiveelementa∈A,thereexistsafinitedimensionalC∗-subalgebraB⊂AwithidB=