p-Operator Space

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1、p-OperatorSpacesZhong-JinRuanUniversityofIllinoisatUrbana-ChampaignGPOTS2008June18-22,20081OperatorSpacesOperatorspacesarespacesofoperatorsonHilbertspaces.AconcreteoperatorspaceisanormclosedsubspaceofB(H)togetherwithamantrixnorm!·!noneachMn(V)givenbyM∼n

2、n(V)⊆Mn(B(H))=B(!2(H)).Theorem[R1988]:LetVbeaBanachspacewithanorm!·!noneachmatrixspaceMn(V).ThenViscompletelyisometricallyisomorphictoaconcreteoperatorspaceifandonlyitsatisfies!"#!!x0!!!M1.!!=max{!x!n,!y!m}!0y!n+mM2.!αxβ!n≤!α!!x!n!β!forallx∈Mn(V),y∈Mm(V)

3、andα,β∈Mn(C)=B(!n).22CompletelyBoundedMapsLetϕ:V→Wbeaboundedlinearmap.Foreachn∈N,wecandefinealinearmapϕn:[xij]∈Mn(V)→[ϕ(xij)]∈Mn(W).Themapϕiscalledcompletelyboundedif!ϕ!cb=sup{!ϕn!:n∈N}<∞.WeletCB(V,W)denotethespaceofallcompletelyboundedmapsfromVintoW,whi

4、chisagainanoperatorspacewithmatrixnormgivenbyMn(CB(V,W))=CB(V,Mn(W)).Inparticular,thedualspaceV∗=CB(V,C)hasanaturaloperatorspacematrixnormgivenbyM∗n(V)=CB(V,Mn(C)).3Inthecategoryofoperatorspaces,itisimportanttoconsiderMatrixNormsandCompletelyBoundedMaps

5、4ExamplesofOperatorSpaces:•C*-algebrasorvonNeumannalgebrasareoperatorspaces.Inparticular,ifGisalocallycompact/discretegroup,C∗(G),C∗(G)andVN(G)λ•DualsA∗ofC*-algebrasandpredualsM∗ofvonNeumannalgebrasA(G)=VN(G)∗,Bλ(G)=C∗(G)∗,andB(G)=C∗(G)∗λ•Moreover,Herz-

6、SchurmultiplieralgebraMcbA(G)⊆CB(A(G),A(G)):McbA(G)={ϕ:G→C:mϕ:ψ∈A(G)→ϕψ∈A(G)with!mϕ!cb<∞}.5ApproximationPropertiesAC*-algebraAissaidtobenuclearifthereexistscomletelycontractivemapsϕα:A→Mn(α)andψα:Mn(α)→Asuchthatψα◦ϕα→idAinthepoint-normtopology.AC*-algeb

7、raAissaidtohavetheCBAP(resp.CCAP)ifthereexistsfiniterankmapsϕα:A→Asuchthat!ϕα!cb≤k(resp.!ϕα!cb≤1)andϕα→idAinthepoint-normtopology.AC*-algebraissaidtohavetheOAPifforeveryx=[xij]∈K(!2)⊗ˇAandε>0,thereexistsafiniterankmapT:A→Asuchthat![T(xij)]−[xij]!K(!2)⊗ˇA<

8、ε.AC*-algebraissaidtobeexactifwehavetheshortexactsequence0→K(!2)⊗ˇA→B(!2)⊗ˇA→Q(!2)⊗ˇA→0,whereQ(H)=B(!2)/K(!2).6LetGbeadiscretegroup.WehavethefollowingimplicationsforgroupC*-algebraC∗(G)λNuclearity⇒CCAP/CBAP⇒OAP⇒Exactness.Ifwecons