Finite cyclic group actions with the tracial Rokhl

arXiv:math/0609785v1[math.OA]28Sep2006FINITECYCLICGROUPACTIONSWITHTHETRACIALROKHLINPROPERTYN.CHRISTOPHERPHILLIPSAbstract.WegiveexamplesofactionsofZ/2ZonAFalgebrasandATalge-braswhichdemonstratethedifferencesbetweenthe(strict)RokhlinpropertyandthetracialRokhlinproperty,andbetween(strict)approximaterepre-sentabilityandtracialapproximaterepresentability.Specificresultsincludethefollowing.WedetermineexactlywhenaproducttypeactionofZ/2ZonaUHFalgebrahasthetracialRokhlinproperty;inparticular,unlikeforthestrictRokhlinproperty,everyUHFalgebraadmitssuchanaction.WeprovethatBlackadar’sactionofZ/2Zonthe2∞UHFalgebra,whosecrossedproductisnotAFbecauseithasnontrivialK1-group,hasthetracialRokhlinproperty,andwegiveanexampleofanactionofZ/2ZonasimpleunitalAFal-gebrawhichhasthetracialRokhlinpropertyandsuchthattheK0-groupofthecrossedproducthastorsion.Inparticular,thecrossedproductofasim-pleunitalAFalgebrabyanactionofZ/2ZwiththetracialRokhlinpropertyneednotbeAF.WegiveexamplesofatraciallyapproximatelyrepresentableactionofZ/2ZonasimpleunitalAFalgebrawhichisnontrivialonK0,andofatraciallyapproximatelyrepresentableactionofZ/2ZonasimpleunitalATalgebrawithrealrankzerowhichisnontrivialonK1.0.IntroductionThetracialRokhlinpropertyforactionsoffinitegroupsonC*-algebraswasin-troducedin[26]forthepurposeofprovingthateverysimplehigherdimensionalnoncommutativetorusisanATalgebra(donein[27]),andprovingthatcertaincrossedproductsofsuchalgebrasbyfinitecyclicgroupsareAFalgebras(donein[5]).Thepurposeofthispaperistoprovideotherexamplesofactionsoffi-nitecyclicgroupswiththetracialRokhlinpropertyonC*-algebraswithtracialrankzero.Wedemonstratebyexamplethedifferencesbetweenthe(strict)RokhlinpropertyandthetracialRokhlinproperty,andbetween(strict)approximaterep-resentability(Definition3.6(2)of[11])anditstracialanalog,tracialapproximaterepresentability(Definition3.2of[26]).(Toemphasizethedistinctionwiththeirtracialanalogs,inthispaperwerefertothestrictRokhlinpropertyandtostrictapproximaterepresentability.)In[4],BlackadarconstructedanactionofZ/2Zonthe2∞UHFalgebrasuchthatthecrossedproducthasnontrivialK1-group,andishencenotAF.Asoneofourexamples,weprovethatthisactionhasthetracialRokhlinproperty.Earlier,inoneoftheexercises(10.11.3)ofhisbook[2],BlackadargaveanexampleofanordertwoautomorphismofK0(A)forasimpleseparableAFalgebraAsuchthat,ifthisautomorphismcouldbeimplementedbyanordertwoautomorphismofA,thentheresultingcrossedproductbyZ/2ZwouldhaveDate:21Sept.2006.2000MathematicsSubjectClassification.Primary46L55;Secondary46L40.ResearchpartiallysupportedbyNSFgrantsDMS0070776andDMS0302401.12N.CHRISTOPHERPHILLIPStorsioninK0ornontrivialK1.WithaveryslightmodificationofBlackadar’salge-bra(weuseZ13⊕ZinsteadofZ12⊕Z),ourexamplesincludeactionsofZ/2ZonthisAFalgebrawiththetracialRokhlinpropertysuchthatK0ofthecrossedproducthasasummandisomorphictoZ/2nZ,andalsoactionswiththetracialRokhlinpropertysuchthatK1ofthecrossedproductisnonzero.Ourresultsgivecounterexamplestovariousstrengtheningsofresultsin[26].Inparticular,usingthenotationZnforZ/nZ:•EvenonaUHFalgebra,anactionofZnwiththetracialRokhlinpropertyneednothavethestrictRokhlinproperty,andinfactthecrossedproductbysuchanactioncanhavenontrivialK1-group,soneednotbeAF.SeeExample3.1,and,forasimpleAFalgebrawhichisnotUHF,Example4.5.•ThecrossedproductofasimpleunitalAFalgebrabyanactionofZnwiththetracialRokhlinpropertycanhavetorsioninitsK0-group—eveniftheactionistraciallyapproximatelyrepresentable.SeeExample4.1.•IfanactionαofZnonasimpleunitalC*-algebraAwithtracialrankzeroisstrictlyapproximatelyrepresentableandhasthetracialRokhlinproperty,itdoesnotfollowthattheautomorphismsofthedualactionareapproximatelyinner—evenifAisUHFandαislocallyrepresentableinthesenseofSectionIof[9].SeeExample2.8.•IfanactionαofZnonasimpleunitalC*-algebraAwithtracialrankzeroistraciallyapproximatelyrepresentableandhasthestrictRokhlinproperty,itdoesnotfollowthatthedualactionhasthestrictRokhlinproperty—evenifAisAT.UsethedualoftheactioninExample3.1.•AtraciallyapproximatelyinnerautomorphismofasimpleunitalC*-algebraAwithtracialrankzeroneednotbetrivialonK0(A)—evenifAisAFandαgeneratesanactionofZnwiththestrictRokhlinproperty.UsethedualoftheactioninExample2.8.•AtraciallyapproximatelyinnerautomorphismofasimpleunitalC*-algebraAwithtracialrankzeroneednotbetrivialonK1(A)—evenifAisATandαgeneratesanactionofZnwiththestrictRokhlinproperty.UsethedualoftheactioninExample3.1.•ThereisanactionαofZnonasimpleAFalgebrasuchthatsuchthatC∗(Zn,A,α)isagainasimpleAFalgebra,butsuchthatαdoesnothavethetracialRokhlinproperty.ThiscanhappenevenwhenthedualactionhasthestrictRokhlinpropertyandαisstrictlyapproximatelyrepresentable,evenlocallyrepresentableinthesenseofSectionIof[9].SeeExample2.9.•ThedualofanactionwiththetracialRokhlinpropertyneednothavethetracialRokhlinproperty,evenwhenboththeoriginalalgebraandthecrossedproductaresimpleAFalgebras,andevenwhentheoriginalac-tioninfacthasthestrictRokhlinproperty.UsethedualoftheactioninExample2.9.ExamplesinvolvingactionsonsimpleC*-algebraswhichdonothavetracialrankzero,manyofthemonsimpleC*-