Tensor products of type III factor representations

arXiv:0805.0667v1[math.OA]6May2008TensorproductsoftypeIIIfactorrepresentationsofCuntz-KriegeralgebrasKatsunoriKawamura∗CollegeofScienceandEngineeringRitsumeikanUniversity,1-1-1NojiHigashi,Kusatsu,Shiga525-8577,JapanAbstractWeintroducedanon-symmetrictensorproductofanytwostatesoranytworepresentationsofCuntz-Kriegeralgebrasassociatedwithacertainnon-cocommutativecomultiplicationinpreviousourwork.Inthispaper,weshowthatacertainsetofKMSstatesisclosedwithrespecttothetensorproduct.Fromthis,weobtainformulaeoftensorproductoftypeIIIfactorrepresentationsofCuntz-KriegeralgebraswhichisdifferentfromresultsofthetensorproductoffactorsoftypeIII.MathematicsSubjectClassifications(2000).46K10,46L30,46L35.Keywords.tensorproductofrepresentations,Cuntz-Kriegeralgebra,KMSstate,typeIIIfactorrepresentation.1IntroductionWehavestudiedstatesandrepresentationsofoperatoralgebras.KMSstatesoverCuntz-Kriegeralgebraswithrespecttocertainone-parameterautomor-phismgroupsareknown[10,13].GNSrepresentationsbysuchKMSstatesinducetypeIIIfactorrepresentationsofCuntz-Kriegeralgebras[26].Ontheotherhand,weintroducedanon-symmetrictensorproductofanytwostatesoranytworepresentationsofCuntz-Kriegeralgebrasassociatedwithacertainnon-cocommutativecomultiplication[25].Fromthese,weinvesti-gatethetensorproductoftwoKMSstatesortheirGNSrepresentationsofCuntz-Kriegeralgebrasinthispaper.Inconsequence,weobtainformulaeoftensorproductoftypeIIIfactorrepresentationsofCuntz-Kriegeralgebras.∗e-mail:[e-mail].1Remarkthatthisisnotastudyofthetensorproductof“factors”oftypeIIIbutthatofacertainnon-symmetrictensorproductoftypeIII“factorrepresentations.”1.1MotivationFromstudiesofbranchinglawsofacertainclassofrepresentationsofCuntzalgebras[20,21,22],wefoundanon-symmetrictensorproductofrepre-sentationsofCuntzalgebras[23].Subsequently,thistensorproductwasexplainedbytheC∗-bialgebrawhichisdefinedbythedirectsumofallCuntzalgebras[24].FurthermoresuchaconstructionofC∗-bialgebrawasgeneralizedtoCuntz-Kriegeralgebras[25].Theessentialtoolofthecon-structionisacertainsetofembeddingsamongCuntz-Kriegeralgebras.Weexplainthisasfollows.AmatrixAisnondegenerateifanycolumnandanyrowarenotzero.For2≤n∞,letMn({0,1})denotethesetofallnondegeneraten×nmatriceswithentries0or1anddefineM≡∪nMn({0,1}).ForA∈M,letOAdenotetheCuntz-KriegeralgebrabyA[6].ForA,B∈M,letA⊠BdenotetheKroneckerproductofAandB[7].ThenMisclosedwithrespectto⊠.Wecanconstructaset{ϕA,B:A,B∈M}suchthatϕA,Bisaunital∗-embeddingofOA⊠BintoOA⊗OBand(ϕA,B⊗idC)◦ϕA⊠B,C=(idA⊗ϕB,C)◦ϕA,B⊠C(A,B,C∈M)(1.1)whereidXdenotestheidentitymapofOXforX=A,CandOA⊗OBmeanstheminimaltensorproductofOAandOB.Wewillgivetheexplicitdefinitionoftheseembeddingsin§1.2.Fromthesetϕ≡{ϕA,B:A,B∈M},wecandefineassociativetensorproductsofstatesorrepresentationsasfollows.LetSAdenotethesetofallstatesoverOA.Forρ1∈SAandρ2∈SB,defineρ1⊗ϕρ2∈SA⊠Bbyρ1⊗ϕρ2≡(ρ1⊗ρ2)◦ϕA,B.(1.2)From(1.1),weseethat(ρ1⊗ϕρ2)⊗ϕρ3=ρ1⊗ϕ(ρ2⊗ϕρ3)(ρ1,ρ2,ρ3∈S∗)whereS∗≡SA∈MSA.Inconsequence,S∗isasemigroupwithrespecttotheoperation⊗ϕ.LetRepOAdenotetheclassofallunital∗-representationsofOA.Asisthecasewithstates,wecandefinetheassociativeoperation⊗ϕontheclassR∗≡SA∈MRepOA:⊗ϕ:R∗×R∗→R∗.2Weshowpropertiesof⊗ϕasfollows:Fortworepresentationsπ1,π2ofaC∗-algebraA,ifπ1andπ2areunitarilyequivalent(resp.quasi-equivalent),thenwewriteπ1≃π2(resp.π1≈π2).(i)Forπi,π′i∈R∗,ifπi≃π′iforeachi=1,2,thenπ1⊗ϕπ2≃π′1⊗ϕπ′2.Fromthis,⊗ϕiswell-definedonthesetR∗≡[A∈M∗({0,1})(RepOA/≃)(1.3)ofallunitaryequivalenceclassesofrepresentationsofOA’s.(ii)Theoperation⊗ϕsatisfiesthedistributionlawwithrespecttothedirectsum.(iii)Theoperation⊗ϕisnon-symmetricinastrongsense,thatis,thereexistA∈Mandπ1,π2∈RepOAsuchthatπ1⊗ϕπ26≃π2⊗ϕπ1.(iv)Evenifbothπ1andπ2areirreducible,π1⊗ϕπ2isnotalwaysirreducible(§4.1of[23]).Inaddition,wecanshowthefollowing.Lemma1.1(i)ThereexistA∈Mandtwofactorrepresentationsπ1andπ2ofOAsuchthatπ1⊗ϕπ2isnotafactorrepresentationofOA⊠A.(ii)ThereexistA∈Mandπ1,π2∈RepOAsuchthatπ1⊗ϕπ26≈π2⊗ϕπ1.(iii)Forπi,π′i∈R∗,ifπi≈π′iforeachi=1,2,thenπ1⊗ϕπ2≈π′1⊗ϕπ′2.(iv)Thereexistρ1,ρ2∈S∗suchthatπρ1⊗ϕπρ26≈πρ1⊗ϕρ2whereπρdenotestheGNSrepresentationbyastateρ.FromLemma1.1(i),evenifbothπ1andπ2arefactorrepresentations,π1⊗ϕπ2isnotalwaysafactorrepresentation.FromLemma1.1(ii),⊗ϕisalsonon-symmetricwithrespecttoquasi-equivalenceclasses.Lemma1.1willbeprovedin§3.4.Ourinterestistocomputetensorproductsofconcretestatesorrep-resentationsofCuntz-Kriegeralgebraswithrespecttotheaboveoperation⊗ϕ.ForthecaseofpermutativerepresentationsofCuntzalgebras,wegaveformulaeofdecompositionoftensorproductscompletely[23].Inthiscase,non-typeIrepresentationneverappear.Inthispaper,weintendtoconsidertensorproductsofnon-typeIrepresentations.31.2AsetofembeddingsofCuntz-KriegeralgebrasInthissubsection,wereviewasetofembeddingsofCuntz-Kriegeralgebras[25].WestatethatamatrixA∈Mn(C)isirreducibleifforanyi,j∈{1,...,n},thereexistsk∈N≡{1,2,3,...}suchthat(Ak)i,j6=0whereAk=A···A(ktimes).For2≤n∞,letMn({0,1})denotethesetofallirreduciblenondegeneraten×nmatriceswithentries0or1,whichisnotapermutationmatrix.DefineM∗({0,1})≡∪{Mn({0,1}):n≥2}.(1.4)ForA=(Aij)∈Mn(C)an