$L^p$-Spaces as Quasi -Algebras

arXiv:funct-an/9410002v15Oct1994Lp-SpacesasQuasi*-AlgebrasF.BagarelloDipartimentodiMatematicaeApplicazioniFacolt`ad’IngegneriaUniversit`adiPalermoI-90128-Palermo-ItalyandC.TrapaniIstitutodiFisicaUniversit`adiPalermoI-90123-Palermo-Italy21.IntroductionLetAbealinearspaceandA′a∗-algebracontainedinA.WesaythatAisaquasi∗-algebrawithdistinguished∗-algebraA′(or,simply,overA′)if(i)therightandleftmultiplicationsofanelementofAandanelementofA′arealwaysdefinedandlinear;and(ii)aninvolution*(whichextendstheinvolutionofA′)isdefinedinAwiththeproperty(AB)∗=B∗A∗wheneverthemultiplicationisdefined.Quasi*-algebras[1,2]ariseinnaturalwayascompletionsoflocallyconvex*-algebraswhosemultiplicationisnotjointlycontinuous;inthiscaseonehastodealwithtopologicalquasi∗-algebras.Aquasi∗-algebra(A,A′)iscalledtopologicalifalocallyconvextopologyτonAisgivensuchthat:(i)theinvolutionA7→A∗iscontinuous(ii)themapsA7→ABandA7→BAarecontinuousforeachB∈A′(iii)A′isdenseinA[τ].Inatopologicalquasi*-algebratheassociativelawholdsinthefol-lowingtwoformulationsA(BC)=(AB)C;B(AC)=(BA)C∀A∈A,∀B,C∈A0Let(X,μ)beameasurespacewithμaBorelmeasureonthelo-callycompactHausdorffspaceX.Asusual,wedenotebyLp(X,dμ)(orsimply,Lp(X)ifnoconfusionispossible)theBanachspaceofall(equivalenceclassesof)measurablefunctionsf:X−→Csuchthatkfkp≡ZX|f|pdμ1/p∞.OnLp(X)weconsiderthenaturalinvolutionf∈Lp(X)7→f∗∈Lp(X)withf∗(x)=f(x).WedenotewithC0(X)theC*-algebraofcontinuousfunctionsvanish-ingatinfinity.Thepair(Lp(X,μ),C0(X))providesthebasiccommutativeexampleoftopologicalquasi∗-algebra.Fromnowon,weassumethatμisapositivemeasure.Inapreviouspaper[3]weintroducedaparticularclassoftopologicalquasi∗-algebras,calledCQ*-algebras.Thedefinitionwewillgivehereisnotthegeneralone,butitisexactlywhatweneedinthecommuta-tivecasewhichwewillconsiderinthispaper.ACQ*-algebraisatopologicalquasi∗-algebra(A,A′)withthefol-lowingproperties3(i)A0isaC*-algebrawithrespecttothenormkk0andtheinvolution*.(ii)AisaBanachspacewithrespecttothenormkkandkA∗k=kAk∀A∈A.(iii)kBk0=max(supkAk≤1kABk,supkAk≤1kBAk)∀B∈A0.Itisshownin[3]thatthecompletionofanyC*-algebra(A0,kk0)withrespecttoaweakernormkk1satisfying(i)kA∗k1=kAk1∀A∈A0(ii)kABk1≤kAk1kBk0∀A,B∈A0isaCQ*-algebrainthesensediscussedabove.Thisisthereasonwhyboth(Lp(X,μ),C0(X))and(Lp(X,μ),L∞(X,μ))areCQ*-algebras.Lp-spacesareexamplesoftheLρ’sconsideredin[4].Letμbeameasureinanon-emptypointsetXandM+bethecollectionofallthepositiveμ−measurablefunctions.Supposethattoeachf∈M+itcorrespondsanumberρ(f)∈[0,∞]suchthat:i)ρ(f)=0ifff=0a.e.inX;ii)ρ(f1+f2)≤ρ(f1)+ρ(f2);iii)ρ(af)=aρ(f)∀a∈R+;iv)letfn∈M+andfn↑fa.e.inX.Thenρ(fn)↑ρ(f).Following[4],wecallρafunctionnorm.LetusdefineLρasthesetofallμ−measurablefunctionssuchthatρ(f)∞.ThespaceLρisaBanachspace,thatisitiscomplete,withrespecttothenormkfk≡ρ(|f|).Ifthefunctionnormρsatisfiestheadditionalconditionρ(|fg|)≤ρ(|f|)kgk∞,∀f,g∈C0(X),thenthecompletionofC0(X)withrespecttothisnormisanabelianCQ*-algebraoverC0(X).Ofcourse,forLp-spaces,kk=kkp.Inthispaperwewilldiscusssomestructurepropertiesof(Lp(X,μ),C0(X))asaCQ*-algebra.InSection2,inparticular,wewillstudyacertainclassofpositivesesquilinearformson(Lp(X,μ),C0(X))whichlead,inrathernaturalway,toadefinitionof*-semisimplicity.Asisshownin[5],*-semisimpleCQ*-algebrasbehavenicelyandforthemarefine-mentofthealgebraicstructureofquasi∗-algebratoapartial*-algebra[6,7]ispossible.TheabeliancaseisdiscussedinSection3.4Finally,wecharacterize*-semisimpleabelianCQ*-algebrasasaCQ*-algebraoffunctionsobtainedbymeansofafamilyofL2-spaces,gener-alizinginthiswaytheconceptofGel’fandtransformforC*-algebras.2.StructurepropertiesofLp-spacesLemma2.1.LetgbeameasurablefunctiononX,withμ(X)∞,andassumethatfg∈Lr(X)forallf∈Lp(X)with1≤r≤p.Theng∈Lq(X)withp−1+q−1=r−1.Proof.LetusconsiderthelinearoperatorTgdefinedinthefollowingway:Tg:f∈Lp(X)−→fg∈Lr(X).Tgisclosed.Indeed,letfnp−→fandTgfnr−→h;thisimpliestheexistenceofasubsequencefnksuchthatfnk−→fa.e.inXandsofnkg−→fga.e.inX.Then,wehavefnkg−→fginmeasure;ontheotherhandthefactthatTgfnr−→himpliesalsothatfngconvergestohinmeasure.Thus,necessarily,h(x)=f(x)g(x)a.e.inX.ThusTgisclosedandeverywheredefinedinLp(X);then,bytheclosedgraphtheorem,Tgisbounded;i.e.,thereexistsC0(dependingong)suchthatkfgkr≤CkfkpIfr=1,thisalreadyimpliesthatg∈Lp′(X)withp−1+p′−1=1.Letnowr1andleth∈Lr′(X)withr−1+r′−1=1Iff∈Lp(X)then,byH¨olderinequality,fh∈Lm(X)withm−1=p−1+r′−1.Thus,thesetF={{h:{∈L√(X),h∈L∇′(X)}isasubsetofLm(X).Conversely,anyfunctionψ∈Lm(X)canbefactorizedastheproductofafunctioninLp(X)andoneinLr′(X).Thisisachievedbyconsidering,forinstance,theprincipalbranchesofthefunctionsψm/pandψm/r′.Wecannowapplythefirstpartoftheprooftoconcludethatg∈Lm′(X)withm−1+m′−1=1.Aneasycomputationshowsnowthatm′=q.Remark2.2.Forr=1andμ(X)notnecessarilyfinite,theabovestatementcanalsobefoundin[4,8].Usingthisfact,withthesametechniqueasabove,thestatementofLemma2.1canbeextendedtothecaseμ(X)=∞.OurproofinvolvesfunctionalaspectsofLp-spacessowethinkisworthgivingit.5Definition2.3.Let(A,A0)beaCQ*-algebra.WedenoteasS(A)thesetofsesquilinearformsΩonA×Awiththefollowingproperties(i)Ω(A,A)≥0∀A∈A(ii)Ω(AB,C)=Ω(B,A∗