Additive maps preserving commutativity up to a factor on nest algebras

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subscription information:Additive maps preserving

commutativity up to a factor on nest

algebras

Meiyan Jiao & Xiaofei Qi

aab Department of Applied Mathematics, Shanxi University of

Finance & Economics, Taiyuan, P.R. China.

b Department of Mathematics, Shanxi University, Taiyuan, P.R.

China.

Published online: 18 Jul 2014.

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Downloaded by [Beijing Institute of Technology] at 18:26 28 March 2015

LinearandMultilinearAlgebra,2015

Vol.63,No.6,1242–1256,http://wendang.chazidian.com/10.1080/03081087.2014.927985

Additivemapspreservingcommutativityuptoafactoronnest

algebras

MeiyanJiaoa?andXiaofeiQib

aDepartmentofAppliedMathematics,ShanxiUniversityofFinance&Economics,Taiyuan,

P.R.China;bDepartmentofMathematics,ShanxiUniversity,Taiyuan,P.R.China

Downloaded by [Beijing Institute of Technology] at 18:26 28 March 2015 CommunicatedbyB.Kuzma(Received30April2014;accepted5May2014)LetNandMbetwonestswithN∈NandM∈McomplementedinNandMwheneverN?=NandM?=M,respectively.Assumethat??:AlgN→AlgMisaunitaladditivesurjection.Inthispaper,alladditivemaps??preservingcommutativityuptoafactorξ(ξ=0,±1)inbothdirections(i.e.??satis?es??(A)??(B)=ξ??(B)??(A)ifandonlyifAB=ξBAforallA,B∈AlgN)arecharacterized.Keywords:nestalgebras;commutativityuptoafactor;JordanisomorphismsAMSSubjectClassi?cations:Primary:47B49;47A121.IntroductionLetAandBbetwoassociativeringsoralgebras.Recallthatanadditivemap??:A→Bpreserveszero-products(inbothdirections)if,forA,B∈A,??(A)??(B)=0whenever(ifandonlyif)AB=0;preservesJordanzero-products(inbothdirections)if,forA,B∈A,??(A)??(B)+??(B)??(A)=0whenever(ifandonlyif)AB+BA=0;preservesLiezero-products(inbothdirections)if,forA,B∈A,??(A)??(B)???(B)??(A)=0whenever(ifandonlyif)AB?BA=0.Thequestionofcharacterizingadditivemapspreservingzero-products(Jordanzero-productsorLiezero-products)hasbeendiscussed,sayin[1–5](seealsoreferencestherein).LetAandBbetwoalgebrasovera?eldF.Forx,y∈Aandξ∈F,ifxy=ξyx,wesaythatxandyarecommutativeuptoafactorξ.Thecommutativityuptoafactor

forpairsofoperatorshasbeenthesubjectofstudyinthecontextofquantumgroups,[6]andtheirmatrixrealizationsgiveexamplesofoperatorpairscommutinguptoafactor.[7]Recallthatanadditivemap??:A→Bpreservescommutativityuptoafactorξinbothdirectionsif,foreveryA,B∈A,??(A)??(B)=ξ??(B)??(A)ifandonlyifAB=ξBA.ItisclearthattheconceptcorrespondstotheabovepreservingLiezero-productsinbothdirections,preservingzero-productsinbothdirectionsandpreservingJordanzero-productsinbothdirectionsifξ=1,0,?1respectively.Cuietal.in[8]characterizedtheunitaladditivesurjectivemapspreservingcommutativityuptoafactorξ(ξ=0,±1)inbothdirectionsbetweenstandardoperatoralgebrasonrealandcomplexin?nite-dimensional?Correspondingauthor.Email:jmyzgl@http://wendang.chazidian.com

©2014Taylor&Francis

LinearandMultilinearAlgebra1243

Banachspaces.Molnár[9]gaveacharacterizationofbijectivelinearmapspreservingcommutativityuptoafactorinbothdirectionsonthealgebraofn×ncomplexmatricesandonthespaceofalln×nself-adjointmatrices,respectively.Here,weremarkthat,in

[9],theconceptofcommutativityuptoafactorisslightlydifferentfromwhatwegive.Theauthor[9]saidthatamap??preservescommutativityuptoafactorinbothdirectionsif,foranyAandB,AB=ξBAforsomeξifandonlyif??(A)??(B)=η??(B)??(A)forsomeη,whereξandηmaybedifferent.Obviously,ourde?nitionismorerestrictive.Thepurposeofthepresentpaperistocharacterizetheadditivemapspreservingcom-mutativityuptoafactorξinbothdirectionsonanotherimportantoperatoralgebra:nestalgebras.Notethatnestalgebrasareneitherself-adjointnorsemi-simpleandprime.Itshouldbepointedoutthat,forthecaseξ=0,HouandZhang[5]characterizedadditivesurjections??with??(I)af?ne(thatis,??(I)injectivewithdenserange)betweennestalgebrasonin?nite-dimensionalBanachspaceswhichpreservezero-productsinbothdirections;forthecaseξ=?1,HouandJiao[3]provedthat,undersomemildconditionsonnests,anunitaladditivesurjectivemap??betweennestalgebraspreservesJordanzero-productsinbothdirectionsifandonlyif??iseitheraringisomorphismoraringanti-isomorphism;forthecaseξ=1,Benkovic?andEremita[10]gaveacompletecharacterizationofalladditivebijectivemapspreservingcommutativityinbothdirectionbetweennestalgebras,undertheadditionalassumptionoftheexistenceofacomplementednontrivialelementinthenest.Hence,wedealwiththecaseξ=0,±1inthispaper.

LetXandYbeBanachspacesovertherealorcomplex?eldF.Asusual,B(X)andF(X)denotetheBanachspaceofallboundedlinearoperatorsfromXintoXandthesubspaceofall?niterankoperatorsinB(X),respectively.AnestonXisachainNofclosed(undernormtopology)subspacesofXcontaining{0}??andX,whichisclosedunderthe??formationofarbitraryclosedlinearspan(denotedby)andintersection(denotedby).AlgNdenotestheassociatednestalgebra,whichisthesetofalloperatorsTinB(X)suchthatTN?NforeveryelementN∈N.WhenN={{0},X},wesaythatNisnontrivial.ItisclearthatifNis.ForN∈N,let????trivial,thenAlgN=B(X)⊥N?={M∈N|M?N},N+={M∈N|N?M}andN?=(N?)⊥,whereN⊥={f∈X?|N?ker(f)}andX?isthedualofX.Let{0}?=0andX+=X.Itiswellknownthatarankoneoperatorx?fbelongstoAlgNifandonlyifthereissome⊥.DenotebyAlgNthesetofall?niterankoperatorsN∈Nsuchthatx∈Nandf∈N?FinAlgN.ItisknownthatAlgFNisadensesubsetofAlgNunderthestrongoperatortopology.Formoreinformationonnestalgebras,wereferto[11].

Thepaperisorganizedasfollows.InSection2,welistthemainresultsobtainedinthepresentpaper.InSection3,wegivesomelemmas,whichareneededforprovingourmainresults.Section4isdevotedtoproofsofthemainresults.

2.Mainresults

Inthissection,wewillstatethemainresultsinthispaper.

Theorem2.1LetNandMbenestsonin?nite-dimensionalBanachspacesXandYovertherealorcomplex?eldF,respectively,withN∈NandM∈McomplementedinXandYrespectivelywheneverN?=NandM?=M.Let??:AlgN→AlgMbeaunitaladditivesurjectivemap.Then??preservescommutativityuptoafactorξinbothdirectionswithξ=0,±1ifandonlyifoneofthefollowingsholds.Downloaded by [Beijing Institute of Technology] at 18:26 28 March 2015

1244

(1)M.JiaoandX.QiIfξ∈R,thenthereexistadimensionpreservingorderisomorphismθ:N→M

andaninvertibleboundedlinearorconjugate-linearoperatorT:X→YsuchthatT(N)=θ(N)foreveryN∈Nand??(A)=TAT?1forallA∈AlgN.Ifξ∈C\Rand|ξ|=1,thenthereexistadimensionpreservingorderisomorphismθ:N→MandaninvertibleboundedlinearoperatorT:X→YsuchthatT(N)=θ(N)foreveryN∈Nand??(A)=TAT?1forallA∈AlgN.

If|ξ|=1,theneitherthereexistadimensionpreservingorderisomorphismθ:N→MandaninvertibleboundedlinearoperatorT:X→YsuchthatT(N)=θ(N)foreveryN∈Nand??(A)=TAT?1forallA∈AlgN,orthereexistadimensionpreservingorderisomorphismθ:N⊥→Mandaninvertible⊥)=θ(N⊥)forboundedconjugate-linearoperatorT:X?→YsuchthatT(N????1everyN∈Nand??(A)=TATforallA∈AlgN.(2)(3)

Downloaded by [Beijing Institute of Technology] at 18:26 28 March 2015 SinceeverylinearsubspaceofaHilbertspaceiscomplemented,thefollowingcorollaryisimmediatefromTheorem2.1.Corollary2.2LetNandMbenestsonin?nite-dimensionalHilbertspacesHandKoverthe(realorcomplex)?eldF,respectively.Let??:AlgN→AlgMbeaunitaladditivesurjectivemap.Then??preservescommutativityuptoafactorξinbothdirectionswithξ=0,±1ifandonlyifoneofthethreeformsinTheorem2.1holds.Forthe?nite-dimensionalcase,itisclearthateverynestalgebraon?nite-dimensionalspacesisisomorphictoanuppertriangularblockmatrixalgebra.LetMn=Mn(F)bethematrixalgebraoverthe?eldF.LetT=T(n1,n2,...,nk)?Mnbeanuppertriangularblockmatrixsubalgebra,i.e.n1+n2+...+nk=n,T={A=(Aij)k×k|Aij∈Mni,njandAij=0ifi>j}.Theorem2.3LetFbetherealorcomplex?eld,andm,nbepositiveintegersgreaterthan1.LetT=T(n1,n2,...,nk)?Mn(F)andS=T(m1,m2,...,mr)?Mm(F)beuppertriangularblockmatrixalgebras,and??:T→Sbeaunitaladditivesurjectivemap.Then??preservescommutativityuptoafactorξinbothdirectionswithξ=0,±1ifandonlyifoneofthefollowingsholds.(1)(n1,n2,...,nk)=(m1,m2,...,mr),thereexistanautomorphismτ:F→Fwithτ(ξ)=ξandaninvertiblematrixT∈Tsuchthat??(A)=TAτT?1forallA∈T;(n1,n2,...,nk)=(mr,mr?1,...,m1),thereexistanautomorphismτ:F→F1andaninvertibleblockmatrixT=(Tij)k×kwithTij∈Mni,njandwithτ(ξ)=?1forallA∈T.Tij=0wheneveri+j>k+1,suchthat??(A)=TAtrτT

HereAτ=(τ(aij))n×nforA=(aij)n×n∈Mn(F)andAtristhetransposeofA.In(1),ifF=R,thenτ=id(i.e.τ(t)=tforallt∈R).(2)

Particularly,if??islinear,wehavethefollowingresults.

Theorem2.4LetFbetherealorcomplex?eld,andm,nbepositiveintegersgreaterthan1.LetT=T(n1,n2,...,nk)?Mn(F)andS=T(m1,m2,...,mr)?Mm(F)beuppertriangularblockmatrixalgebras.Assumethat??:T→Sisalinearsurjective