Tracial Rokhlin property for automorphisms on simp

arXiv:math/0601513v1[math.OA]20Jan2006TracialRokhlinpropertyforautomorphismsonsimpleAT-algebrasHuaxinLinDepartmentofMathematicsUniversityofOregonEugene,Oregon97403-1222HiroyukiOsakaDepartmentofMathematicalSciencesRitsumeikanUniversityKusatsu,Shiga,525-8577,JapanAbstractLetAbeaunitalsimpleAT-algebraofrealrankzero.Givenanisomorphismγ1:K1(A)→K1(A),weshowthatthereisanautomorphismα:A→Asuchthatα∗1=γ1whichhasthetracialRokhlinproperty.Consequently,thecrossedproductA⋊αZisasimpleunitalAH-algebrawithrealrankzero.WealsoshowthatautomorphismwithRokhlinpropertycanbeconstructedfromminimalhomeomorphismsonaconnectedcompactmetricspace.1IntroductionAlanConnesintroducedtheRokhlinpropertyinergodictheorytooperatoralgebras([2]).SeveralversionsoftheRokhlinpropertyforautomorphismsonC∗-algebrashavebeenstudied(forexample[8],[32],[14],[27],[11],[12]and[30],tonameafew).GivenaunitalC∗-algebraAandanautomorphismαonA,onemayviewthepair(A,α)asanon-commutativedynamicalsystem.Tostudyitsdynamicalstructure,itisnaturaltointroducethenotionofRokhlinproperty.LetAbeaunitalsimpleAT-algebra(aC∗-algebrawhichisaninductivelimitofthoseC∗-algebrasthatarefinitedirectsumsofcontinuousfunctionsonthecircleT)andletαbeanautomorphismonA.Inseveralcases,Kishimotoshowedthatifαisapproximatelyinner(orhomotopictotheidentity)andhasaRokhlintypeproperty,thenthecrossedproductA⋊αZisagainaunitalsimpleATofrealrankzero([14],[15],[16]).ItisprovedthatifαinducestheidentityonK0(A)(ora“dense”subgroupofK0(A))andαhasso-calledtracialcyclicRokhlinpropertythenindeedthecrossedproductisanAH-algebraofrealrankzero([25]and[24]).AnaturalquestioniswhenautomorphismshavecertainRokhlinproperty.In[29],itisshownthatifAisaunitalseparablesimpleC∗-algebrawithtracialrankzeroandwithauniquetracialstateandifαisanautomorphismsuchthatA⋊αZhasaunqiuetrace,thenαhasthetracialRokhlinproperty.Morerecently,N.C.Phillips[31]showedthat,foranyunitalsimpleseparableC∗-algebraAwithtracialrankzero,thereisadenseG-δsetofapproximatelyinnerautomorphismssuchthateveryautomorphisminthethesethasthetracialRokhlinproperty.1LetAbeaunitalsimpleAT-algebraofrealrankzero.Supposethatγ1:K1(A)→K1(A)isanautomorphism.Inthisnote,wepresentanautomorphismαonAwiththetracialcyclicRokhlinpropertysuchthatα∗1=γ1.Theautomorphismαthatwepresentinthisnotealsohasthepropertythatα∗0=idK0(A).WhenαhasthetracialcyclicRokhlinproperty,thenαmustfixalargesubgroupofK0(A).Infact,itisshownin[24]that,atleastinthecasethatAhasauniquetracialstate,ifαhastracialcyclicRokhlinproperty,thenα2fixesasubgroupG⊂K0(A)sothatρ(G)isdenseinAff(T(A)),whereT(A)isthetracialstatespaceofA,Aff(T(A))isthespaceofallrealaffinecontinuousfunctionsonT(A)andρ:K0(A)→Aff(T(A))isthepositivehomomorphisminducedbytheevaluationρ([p])(τ)=τ(p)forprojectionsp∈A.LetXbeaconnectedcompactmetricspaceandρX:K0(C(X))→Zbethedimensionmap.DenotebykerρXthekernelofthedimensionmap.GivenasuchXandacountabledensesubgroupD⊂Q,thereisastandardwaytoconstructaunitalsimpleC∗-algebraAXwithtracialrankzerosuchthatK0(AX)=D⊕kerρXandK1(AX)=K1(C(X))aswellasthereisaunitalembeddingj:C(X)→AXsothat(j∗0)|kerρX=id|kerρXandj∗1=idK1(X).Thiscouldbeviewedasaversionofthenon-commutativespaceassociatedwithX.Supposethatψ:X→XisaminimalhomeomorphismonX.WeshowthatonecanconstructanautomorphismαonAXassociatedwithψsuchthatαhasthetracialcyclicRokhlinpropertyandsuchthatα∗0|D=idD,α∗0|kerρX=ψ∗0|kerρXandα∗1=ψ∗1.Thismaybeviewedasanon-commutativeversionoftheminimalactionassociatedwithψ.Wealsoshowthatsomewhatgeneralconstructioncanalsobemade.ItappearsthatautomorphismswiththetracialcyclicRokhlinpropertyoccurquiteoften.WhilewebelievethatmanyothertypesofconstructionofautomorphismswiththeRokhlinpropertymaybepossibleandperhapsnotnecessarilydifficult,wethinktheseconstructioninthisnoteshedsomelightonhowcommutativeRokhlintowerlemmaappearsnaturallyinthestudyofnon-commutativedynamicalsystemssuchas(A,α).AknowledgementThefirstnamedauthorwaspartiallysupportedbyaNSFgrant.ThesecondauthorwaspartiallysupportedbyOpenResearchCenterProjectforPrivateUniversi-ties:machingfundfromMEXT,2004-2008.MuchofthegroundworkofthisreseachwasdonewhenbothauthorswerevisitingEastChinaNormalUniversityinthesummer2004.TheywouldliketoacknowledgethesupportfromShanghaiPriorityAcademicDisciplinesandfromDepartmentofMathematicsofEastChinaNormalUniversity.2PreliminariesThefollowingconventionswillbeusedinthispaper.(1)LetAbeastablyfiniteC∗-algebra.DenotebyT(A)thetracialstatespaceofA.DenotebyAff(T(A))thenormedspaceofallrealaffinecontinuousfunctionsonT(A).(2)DenotebyρA:K0(A)→Aff(T(A))thepositivehomomorphisminducedbyρA([p])(τ)=τ⊗Tr(p)foranyprojectioninMk(A)(k=1,2,...,),whereTristhestandardtraceonMk.(3)LetXbeaconnectedcompactmetricspace.DenotebyρX:K0(C(X))→ZthepositivehomomorphismρC(X)definedin(2).ItisthedimensionfunctionfromK0(C(X))toZ.(4)LetXbeacompactmetricspace,letFbeasubsetofXandletε0.PutFε={x∈X:dist(x,F)ε}.2(5)LetAbeaC∗-algebraandletpandqbeprojectionsinA.Wesaypisequivalenttoqifthereisav∈Asuchthatv∗v=pandvv∗=q.(6)LetA=limn→∞(An,φn)beaninductivelimitofC∗-algebras.HereφnisahomomorphismfromAnintoAn+1.Wewilluseφn,∞:An→Aforthehomom