XI CUI AND JI GUo-XING

(College of Mathematics and Information Science,Shaanxi Normal University,Xi’an,710062)

Communicated by Ji You-qing

Additive Maps Preserving the Star Partial Order onB(H)

XI CUI AND JI GUo-XING*

(College of Mathematics and Information Science,Shaanxi Normal University,Xi’an,710062)

Communicated by Ji You-qing

LetB(H)be theC?-algebra of all bounded linear operators on a complex Hilbert spaceH.It is proved that an additive surjective mapφonB(H)preserving the star partial order in both directions if and only if one of the following assertions holds.(1)There exist a nonzero complex numberαand two unitary operatorsUandVonHsuch thatφ(X)=αUXVorφ(X)=αUX?Vfor allX∈B(H).(2) There exist a nonzeroαand two anti-unitary operatorsUandVonHsuch thatφ(X)=αUXVorφ(X)=αUX?Vfor allX∈B(H).

linear operator,star partial order,additive map

1 Introduction

In the last few decades,many researchers have studied properties of various partial orders on matrix algebras,or operator algebras acting on a complex infnite dimensional Hilbert space,such as minus partial order,star partial order,left and right star partial order and so on(see[1-6]).One of the orders on the algebraMnof alln×ncomplex matrices is the star partial orderdefned by Drazin in[5].LetA,B∈Mn.Then we say thatandWe note that this defnition can be extended to aC?-algebra by the same way.In particular,it can be extended to theC?-algebraB(H)of all bounded linear operators on a complex Hilbert spaceH.For example,motivated byˇSemrl’s approach presented in[7]for minus partial order,Dolinar and Marovt[4]gave an equivalentdefnition(see Defnition 2 in[4])of the star partial order and considered some properties of this partial order.We can refer[1,4]to see more interesting properties.

On the other hand,as partially ordered algebraic structures onMnandB(H),what are the automorphisms ofMnandB(H)with respect to those partial orders?These topics have been studied and some interesting results have been obtained.ˇSemrl[7]described the structure of corresponding automorphisms for the minus partial order.For the star partial order,Guterman[8]characterized linear bijective maps onMnpreserving the star partial order and Legiˇsa[9]considered automorphisms ofMnwith respect to the star partial order. Recently,several authors consider the automorphisms of certain subspaces ofB(H)with respect to the star partial order whenHis infnite dimensional.Dolinar and Guterman[10]studied the automorphisms of the algebraK(H)of compact operators on a separable complex Hilbert spaceHand they characterized the bijective,additive,continuous maps onK(H) which preserve the star partial order in both directions.On the other hand,characterizations of certain continuous bijections on the normal elements of a von Neumann algebra preserving the star partial order in both directions are obtained by Bohata and Hamhalter[11].In this paper,we consider additive surjective maps preserving the star partial order in both directions onB(H)and characterizations of those maps are given.In particular,we improve the main result in[10].

LetHbe a complex Hilbert space and denote by dimHthe dimension ofH.Let C and Q denote the complex feld and the rational number feld,respectively.LetB(H),K(H) andF(H)be the algebras of all bounded linear operators,the compact operators and the fnite rank operators onH,respectively.For every pair of vectorsx,y∈H,〈x,y〉denotes the inner product ofxandy,andx?ystands for the rank-1 linear operator onHdefned by(x?y)z=〈z,y〉xfor anyz∈H.Ifxis a unit vector,thenx?xis a rank-1 projection.σ(A)is the spectrum ofAfor anyA∈B(H).For a subsetSofH,[S]denotes the closed subspace ofHspanned bySandPMdenotes the orthogonal projection onMfor a closed subspaceMofH.We denote byR(T)andN(T)the range and the kernel of a linear mapTbetween two linear spaces.Throughout this paper,we generally denote byIthe identity operator on a Hilbert space.

2 Additive Maps Preserving the Star Partial Order

Letφbe an additive map onB(H).We say thatφpreserves the star partial order iffor anyA,B∈B(H)such thatWe say thatφpreserves the star partial order in both directions ifif and only iffor anyA,B∈B(H). We frstly give the following lemma which generalizes Lemma 10 in[10].

LetT∈B(H).We denote by

respectively.Then

and

with respect to the orthogonal decompositions(2.1),whereT0∈B(H1,K1)is an injective operator with dense range.

Lemma 2.1LetT∈B(H)be a nonzero operator.ThenTis of rank-1if and only if for any operatorSwithwe haveS=0orS=T.

PutU1=U|E(Δ)H1,A1=E(Δ)A,U2=U|(I-E(Δ))H1andA2=(I-E(Δ))AonH1, respectively.Then

according to(2.3).Let

Theorem 2.1Let φ be an additive surjective map on B(H).Then φ preserves the star partial order in both directions if and only if one of the following assertions hold:

(1)There exist a nonzero α∈Cand two unitary operatorsUandVon H such that φ(X)=αUXVor φ(X)=αUX?Vfor allX∈B(H);

(2)There exist a nonzero α∈Cand two anti-unitary operatorsUandVon H such that φ(X)=αUXVor φ(X)=αUX?Vfor allX∈B(H).

Proof.The sufciency is clear.We only need prove the necessity.It is clear thatφis injective.Thenφ-1preserves the star partial order too.We complete the proof by several steps.

Step 1.φpreserves rank-noperators in both directions.

LetAbe a rank-1 operator andφ(A)=B.Suppose that rankB≥2.Then there is a nonzeroB1∈B(H)such thatB1?≤BandB1/=Bby Lemma 2.1.PutA1=φ-1(B1).

ThusBis of rank-1.It follows thatφpreserves rank-1 operators in both directions.Since a rank-noperator is the sum ofnrank-1 operators,it is elementary thatφpreserves rank-noperators in both directions.

Step 2.Letf,g∈Handφ(f?g)=u?v.Then

In fact,for anyx∈{f}⊥,y∈{g}⊥,we have

Letφ(x?y)=ξ?η.Then

which implies that

The converse is the same sinceφpreserves the star partial order in both directions.

Step 3.For any unit vectorsMoreover,iff⊥g, then

LetandBy Step 2,ξ1⊥ξ2andη1⊥η2. Without loss of generality,we may assume that

PutUandVbe two unitary operators onHsuch that

Letψ=UφV?.Thenψpreserves the star partial order in both directions such that

and

Thenψpreserves rank-1 operators in both directions.LetNote that bothandare of rank-1.Then eitherf(resp.g)andξ3orf(resp.g)andη3are linearly dependent.We assume that

We thus have

Put

ThenE(r)is a projection and

Of course,

It follows that

Then

Thus

If dimH=2,then for any unit vectorx∈Hwe have

Thus we haveψ(x?x)is a projection and

Assume that dimH＞2.For any unit vectorsfandg,take any unit vectoThen

We next assume that‖φ(f?f)‖=1 for any unit vectorf∈Hwithout loss of generality.

Step 4.Let{eλ:λ∈Λ}be an orthonormal basis ofH.Then there are two orthonormal bases{fλ:λ∈Λ}and{gλ:λ∈Λ}such that

If(2.4)holds,then both{fλ:λ∈Λ}and{gλ:λ∈Λ}are orthonormal families ofH.If there is a unit vectorf∈Hsuch thatf⊥fλfor allλ∈Λ,thenφ-1(f?gλ0)=x0?y0is a rank-1 operator.By Step 2,eλ∈{x0}⊥.This is a contradiction.Thus both{fλ:λ∈Λ}and{gλ:λ∈Λ}are bases ofH.

Step 5.φis linear or conjugate linear onF(H).

As in Step 4,let{eλ:λ∈Λ}be an orthonormal basis ofH.LetUandVbe two unitary operators onHsuch thatU1fλ=eλandV1gλ=eλfor anyλ∈Λ.Put

Thenφ1preserves the star partial order in both directions such that

For anyn∈N+and{eλi:1≤i≤n}?{eλ:λ∈Λ},denote

We conclude that

by the similar way as Step 4 of[10].In fact,it easily follows thatφ1(Q)=Q,whereQis the projection onto{eλ:λ∈S}for any subsetS?Λ.For anyA∈PnB(H)Pn,we know that

Then

It follows thatby a simple calculation.PnB(H)Pncan be identifed withMn.Socan be considered as a bijective,additive map onMn,which preserves the star partial order in both directions.It follows from Theorem 3.1 in[12] thatis linear or conjugate linear.We note that ifis linear(resp. conjugate linear)for somek≥2,thenis also linear(resp.conjugate linear)for anyn.This implies that ifis linear(resp.conjugate linear)for somek≥2,thenis linear(resp.conjugate linear)for anyn.We now assume thatis linear for somek≥2.LetLetMbe the subspace generated by

ThenMis fnite dimensional with an orthonormal basis{hj:1≤j≤m}containingIt now follows thatis linear by preceding proof since

Thusφis linear onF(H).

Ifis conjugate linear for somek≥2,thenφis conjugate linear onF(H).

We now next assume thatφis linear onF(H).Thenφis a rank preserving linear bijection onF(H).It follows from Theorem 2.1.6 in[13]that the following statements hold.

(1)There exist two linear mapsAandConHsuch that for allx,y∈H,

(2)There exist two conjugate linear mapsAandConH,such that for allx,y∈H,

Note that bothAandCare invertible sinceφis bijective onF(H).Assume that(1) holds.Then for any unit vectorse,f∈Hsuch that〈e,f〉=0,we have that

by Step 2.Note that(e+f)⊥(e-f).It follows that

If dimH=2,then for any unit vectorx∈H,we havex=αe+βffor some constantsα,β∈C with|α|2+|β|2=1.We easily have that

by an elementary calculus.If dimH＞2,then for any unit vectorsx,y∈H,there is a unit vectorz∈{x,y}⊥.It now follows that

Put

Then?is an additive bijection onB(H)preserving the star partial order in both directions such that

Now letP∈B(H)be any projection.Then for any fnite rank projectionQ,ifQ≤P,we have

Then

Noting that{λQ:Q≤P}is a?-increasing net and?-bounded from above such that

in strong operator topology,by Proposition 3.5 in[1],we have

We note that the?-increasing and?-bounded sequences are considered in this proposition. However,the proposition still holds if we replaces a sequence by a net.By considering?-1, we have

Then?(X)=Xfor allX∈B(H)sinceXis a linear combination of fnitely many projections from Theorem 3 in[14].Thus

If(2)holds,then there are two anti-unitary operatorsUandVsuch that

Ifφis conjugate linear onF(H),then we similarly have two unitary operatorsUandVonHsuch that

or two anti-unitary operatorsUandVonHsuch that

The proof is completed.

The following corollary is a generalization of the main result in[10].

Corollary 2.1Let φ be an additive surjective map on K(H).Then φ preserves the star partial order in both directions if and only if one of the following holds:

(1)There exist a nonzero α∈Cand two unitary operatorsUandVon H such that

or

(2)There exist a nonzero α∈Cand two anti-unitary operatorsUandVon H such that

or

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A

1674-5647(2015)01-0089-08

10.13447/j.1674-5647.2015.01.10

Received date:July 11,2013.

Foundation item:The NSF(11371233)of China and the Fundamental Research Funds(GK201301007)for the Central Universities.

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E-mail address:xicui@snnu.edu.cn(Xi C),gxji@snnu.edu.cn(Ji G X).

2010 MR subject classifcation:47B49,47B47