On the structure of tensor products of l_p spaces

arXiv:math/9402205v1[math.FA]8Feb1994ONTHESTRUCTUREOFTENSORPRODUCTSOFℓp-SPACESbyAlvaroArias∗TheUniversityofTexasatSanAntonioandJeffD.Farmer∗∗TheUniversityofNorthernColoradoAbstract.Weexaminesomestructuralpropertiesof(injectiveandprojective)tensorproductsofℓp-spaces(projections,complementedsubspaces,reflexivity,isomorphisms,etc.).Wecombinetheseresultswithcombinatorialargumentstoaddressthequestionofprimarityforthesespacesandtheirduals.∗ThisresearchwaspartiallysupportedbyNSFDMS–8921369.∗∗Portionsofthispaperformapartofthesecondauthor’sPh.D.dissertationunderthesupervisionofW.B.Johnson.ThisresearchwasalsopartiallysupportedbyNSFDMS–8921369.00.Introduction.ABanachspaceXisprimeifeveryinfinite-dimensionalcomplementedsubspacecon-tainsafurthersubspacewhichisisomorphictoX.ABanachspaceXissaidtobeprimaryifwheneverX=Y⊕Z,XisisomorphictoeitherYorZ.Theclassicalexamplesofprimespacesarethespacesℓp,1≤p≤∞.Manyspacesderivedfromtheℓp-spacesinvariouswaysareprimary(seeforexample[AEO]and[CL]).TheprimarityofB(H)wasshownbyBlower[B]in1990,andArias[A]hasrecentlydevelopedfurthertechniqueswhichareusedtoprovetheprimarityofc1,thespaceoftraceclassoperators(thiswasfirstshownbyArazy[Ar1,Ar2]).IthasbecomeclearthatthesetechniquesarenotnaturallyconfinedtoaHilbertspacecontext;inthepresentpaperwewishtoextendtheresultstoavarietyoftensorproductsandoperatorspacesofℓp-spaces(andinsomecasesLp-spaces).Wealsoincludesomerelatedresults.Someoftheintermediatepropositions(onfactoringoperatorsthroughtheidentity)mayactuallybetrueforawiderclassofBanachspaces(thosewithunconditionalbaseswhichhavenontriviallowerandupperestimates).Infact,thecombinatorialaspectsofthefactorizationcanbeappliedquitegenerally,andmayhaveotherapplications.Theproofsofprimarity,however,relyonPelczy´nski’sdecompositionmethodwhichisnotsoreadilyextended.Wehavethuskeptmainlytothecaseofinjectiveandprojectivetensorproductsofℓpspacesthroughout.TheresultsweobtainapplytothegrowingstudyofpolynomialsonBanachspacessincepolynomialsmaybeconsideredassymmetricmultilinearoperatorswithanequivalentnorm(see[FJ],[M],or[R]).Ourmainresultsare:(1)If1p∞,thenB(ℓp)≈B(Lp).(2)If1pi+1pj≤1foreveryi6=j,orifallofthepi’sareequal,thenℓp1ˆ⊗···ˆ⊗ℓpNisprimary.(3)ℓpembedsintoℓp1ˆ⊗···ˆ⊗ℓpNifandonlyifthereexistsA⊂{1,2,···,n}suchthat1p=min{Pi∈A1pi,1}.(4)If1≤p∞andm≥1,thenthespaceofhomogeneousanalyticpolynomialsPm(ℓp)andthesymmetrictensorproductofmcopiesofℓpareprimary.1Thepaperisorganizedasfollows.InSection1wesetnotation,definitionsandsomenecessarybutmoreorlessknownfacts.InSection2weshowthatB(ℓp),theBanachspaceofboundedlinearoperatorsonℓp,isisomorphictoB(Lp),andinfacttoB(X)wheneverXisaseparableLp-space,alongwithsomemoregeneralresultswerequirelater.InSection3wewillconstructamultiplierthroughwhichagivenoperatorontensorproductsmaybefactored;wethenusethistoshowthatsomeprojectivetensorproductsareprimary.InSection4wewillprovethattheℓpsubspacesofℓp1ˆ⊗···ˆ⊗ℓpNarethe“obvious”onesandusethistoprovethatsomeprojectivetensorproductsarenotprimary(forexample,ℓ2ˆ⊗ℓ1.5isnotprimary).Section5coversthequestionofprimarityintheinjectivetensorproductsandoperatorspaces,asituationnotalwaysdualtotheprojectivecaseandcallingforsomewhatdifferenttechniques.Section6isanappendixinwhichweprovethetechnicallemmasweuseinSection3.WewouldliketothankW.B.JohnsonfororganizingthesummerworkshopsinLinearAnalysisandProbabilityatTexasA&MUniversityin1991-1993,andtheNSFforfundingthem.1.Preliminaries.Unlessexplicitlystated,allreferencestoℓpspaceswillassumethat1p∞,andwilladherethenotationalconventionthat1pi+1qi=1orsometimes1r+1r′=1.DefineX=ℓp1ˆ⊗···ˆ⊗ℓpN.WecanidentifyitspredualX∗anddualX∗asfollowsX∗=ℓq1ˇ⊗···ˇ⊗ℓqNX∗=B(ℓp1,(ℓp2ˆ⊗···ˆ⊗ℓpN)∗)≡B(ℓp1,B(···B(ℓpN−1,ℓqN)···)).TheelementsofX,X∗,orX∗haverepresentationsasaninfiniteN-dimensionalmatrixofcomplexnumbers(wemustkeepinmind,however,thatthisrepresentationmaynotbethemostefficientforcomputingthetensorproductnorm)wheretheelementintheα=(α1,···,αN)∈NNpositionisthecoefficientofthe“matrixelement”eα=eα1⊗···⊗eαN2witheαjbeingtheαj-thelementintheunitvectorbasisofℓpj.Allsubspacesweconsiderarenorm-closed,andwhenweindicatethelinearspanofelementswealwaysmeantheclosedspan.Thefollowingelementarylemmaisveryimportanttothestructureofprojectivetensorproducts.LEMMA1.1.LetXandYbeBanachspacesandS∈B(X),T∈B(Y).ThenS⊗T∈B(Xˆ⊗Y)isdefinedbyS⊗T(x⊗y)=S(x)⊗T(y)andsatisfieskS⊗Tk≤kSkkTk.AsaconsequenceofthiswegetthatprojectivetensorproductsofBanachspaceswithbaseshavebases.PROPOSITION1.2.LetXandYbeBanachspaceswithbases(en)nand(fn)nrespectively.ThenXˆ⊗Yhasabasis.Moreover,wetaketheelementsofthebasisfromthe“shell”∂Mn=[ei⊗ej:max{i,j}=n];i.e.,e1⊗f1,e2⊗f1,e2⊗f2,e1⊗f2,e3⊗f1,e3⊗f2,e3⊗f3,e2⊗f3,e1⊗f3,···,etc.Theproofofthisiseasy.OntheonehanditisclearthatthespanofthosevectorsisdenseandusingLemma1.1(withtheoperatorsreplacedbyprojections)weseethattheinitialsegmentsareuniformlycomplemented,because∂Mnisclearlycomplemented.Asaconsequencewegetthatℓp1ˆ⊗···ˆ⊗ℓpNhasabasisconsistingofeα’s.Moreover,wecanuseLemma1.1toprovethat∂Mn=[eα:α∈NN,max{α1,···,αN}=n]is2-complementedandthat(∂Mn)nformsaSchauderdecomposition