on certain extension properties for the space of compact operators

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1、ONCERTAINEXTENSIONPROPERTIESFORTHESPACEOFCOMPACTOPERATORSTimurOikhbergandHaskellP.RosenthalAbstract.LetZbeafixedseparableoperatorspace,X⊂Ygeneralseparableoperatorspaces,andT:X→Zacompletelyboundedmap.ZissaidtohavetheCompleteSeparableExtensionProperty(CSEP)ifeverysuchmapadmitsacompletelyboundedex

2、tensiontoY;theMixedSeparableExtensionProperty(MSEP)ifeverysuchTadmitsaboundedextensiontoY.Finally,ZissaidtohavetheCompleteSeparableComplementationProperty(CSCP)ifZislocallyreflexiveandTadmitsacompletelyboundedextensiontoYprovidedYislocallyreflexiveandTisacompletesurjectiveisomorphism.LetKdenotet

3、hespaceofcompactoperatorsonseparableHilbertspaceandK0thec0sumofMn’s(thespaceof“smallcompactoperators”).ItisprovedthatKhastheCSCP,usingthesecondauthor’spreviousresultthatK0hasthisproperty.Anewproofisgivenfortheresult(duetoE.Kirchberg)thatK0(andhenceK)failstheCSEP.ItremainsanopenquestionifKhasth

4、eMSEP;itisprovedthisisequivalenttowhetherK0hasthisproperty.AnewBanachspaceconcept,ExtendableLocalReflexivity(ELR),isintroducedtostudythisproblem.Furthercomplementsandopenproblemsarediscussed.ContentsIntroduction...............................................................................2§1.E

5、xtendingCompleteIsomorphismsintoB(H)...............................................4§2.AnoperatorspaceconstructiononcertainsubspacesofM∞...............................11§3.Theλ-MixedSeparableExtensionPropertyandExtendablyLocallyReflexiveBanachspaces..............................................

6、.....................17§4.K0failstheCSEP:anewproofandgeneralizations.......................................33References................................................................................40arXiv:math/9905017v1[math.OA]4May1999TypesetbyAMS-TEX12TIMUROIKHBERGANDHASKELLP.ROSENTHALIntr

7、oductionThespaceKofcompactoperatorsonaseparableinfinitedimensionHilbertspaceHisoftenthatthoughtofasthenon-commutativeanalogueofc0,thespaceofsequencesvanishingatinfinity.Indeed,ifoneregardsKasmatriceswithrespecttoafixedorthonormalbasisofH,t