Centralizers And Lie Derivable Maps On Operator Algebras

As is well known,left(right)centralizers,centralizers and Lie derivations are very im-portant maps on operator algebras and operator theory,and have received a fair amount of attention.In this paper,we characterize additive maps on triangular rings、prime rings and von Neumann algebras which are centralized at one given point,we characterize additive maps on B(X)which are derivable at a operator with ker(Ω)≠ 0 or ran(Ω)≠ H.The structure of this paper is as follows:In the first chapter,we introduce the background of the discussed problem of this thesis briefly,main results,and notations and basic theorems in this thesis.In the second chapter,we give new equivalent characterization of centralizers.Main results are as follows:1.Equivalent characterization of centralizers on triangular rings、prime rings and von Neumann algebras are obtained.Let T = Tri(A,M,B)be a triangular rings.Assume that,for each A∈A,B ∈B,there are some integers n1,n2 such that n1I1—A and n2I2-B are invertible in A and B respectively.Then an additive map Φ:T → T is centralized at an arbitrary but fixed element Z E T if and only if it is a centralizer.2.Equivalent characterization of centralizers on prime rings is obtained.Let R be a,prime rings having the unit I and a nontrivial idempotent P.Assume that,for each A11∈ R11,there is an integer n such that nP1-A11 is invertible in R11 Then an additive map Φ:R→R is centralized at Z ∈R with Z = PZ,that is,Φ(AB)= Φ(A)B = AΦ(B)for any A,B ∈R with AB = Z if and only if Φ(AB)= Φ(A)B = AΦ(B),VA,B ∈R.3.Equivalent characterization of centralizers on von Neumann algebras is obtained.Let M be a von Neumann algebra without central summands of type I1,and let Z ∈M be such that(I-P)Z = 0 for a projection P ∈ M with P = O and P = I.Then an additive mapΦ:M→M satisfies Φ(AB)= Φ(A)B = AΦ(B)for any A,B ∈M with AB = Z if and only if Φ(AB)= Φ(A)B = AΦ(B)for all A,B ∈M.In the third chapter,we characterize Lie derivations on B(X).Let X be a Banach space with dimension at least 2 and δ:B(X)→ B(X)be an additive map.In this paper,we show that If Ω ∈ B(X)such that PΩ＝ Ω for a nontrivial idempotent P ∈ B(X),then δ is Lie derivable at Ω which are δ([A,B])=[δ(A),B]+[A,δ(B)]for any A,B ∈ B(X)with AB =Ω if and only if there exists a derivation T:B(X)→B(X)and an additive map f:B(X)→ IF vanishing at commutators[A,B]with AB =Ω such thatδ(A)=-T(A)+f(A)I,(?)A ∈ B(X).In particular,if X = H is a,Hilbert space,Ω∈B(H)such that ker(Ω)≠0 or ran(Ω)≠H,then δ is Lie derivable at Ω if and only if it has the above form.