Studying Some Mappings On Operator Algebras

In this paper,we discuss of some mappings on operator algebras.The mappings that we study include derivations,inner derivations,2-local derivations,Jordan deriv- able mappings and Jordan homomorphisms.The algebras that we study include matrix algebras,standard operator algebras,von Neumann algebras,C*-algebras,semisimple Banach algebrs,generalized matrix algebras,the algebras of locally measurable operators affiliated with a von Neumann algebra,subspace lattice algebras,and so on.This paper splits into six chapters.In Chapter one,we first introduce the background of this study.Then we give the problems that we would discuss and review the relevant developments and achievements until now.In the last,we give the definitions of the algebras and the mappings that we would mention in this paper.In Chapter two,we discuss derivations and inner derivations on matrix algebras and algebras of locally measurable operators affiliated with a finite type I von Neumann algebra R.Let A be a unital algebra over C and M be a unital A-bimodule.We show that every derivation D:Mn(A)? Mn(M),n? 2,can be represented as a sum D=Dm+?,where Dm is an inner derivation and ? is a derivation induced by a derivation ? from A into M.In addition,the representation of the above form is unique if and only if A commutes with M.Let R be a finite von Neumann algebra of type I with center Z and LS(R)be the algebra of locally measurable operators affiliated with R.We also prove that if the lattice ZP of all projections in Z is atomic,then every derivation D:R?LS(R)is an inner derivation.In Chapter three,we discuss 2-local inner derivations and 2-local derivations from Mn(A)into Mn(M).Let A be an algebra and let M be an A-bimodule.Then M is symmetric if ax=xa(a ? A,X?M).For the case that M is symmetric,we obtain thatevery 2-local inner derivation from Mn(A)into Mn(M)is an inner derivation.In addition, if A is commutative,we prove that every 2-local derivation ?:Mn(A)?Mn(M),n? 2, is a derivation.Let R be an arbitrary von Neumann algebra without abelian direct summands.We also show every 2-local derivation ?:R?LS(R)is a derivation.In Chapter four,we study 2-local derivations on semisimple Banach algebras.Let A be a semisimple Banach algebra with minimal left ideal.Then the smallest ideal contained all minimal left ideals is called the socle of A and is denoted by soc(A).We prove that if the closure of soc(A)is an essential ideal of A,then every 2-local derivation on A is a derivation.We also show that every 2-local derivations on some operator algebras,such as standard operator algebras,semisimple modular annihilator Banach algebras,group algebras,strongly double triangle subspace lattice algebras and J-subspace lattice algebras,is a derivation.In Chapter five,we discuss Jordan derivable mappings on generalized matrix algebras through commutative zero products.Let u be a generalized matrix algebra.We prove that if ?:u?u is a linear mapping such that ?(U)(?)V+U(?)?(V)=0 whenever UV=VU=0,then ?=?+?,where ? is a Jordan derivation and 77 is a multiplier.We also prove that the similar conclusion remains valid on full matrix algebras,completely distributive commutative subspace lattice algebras,triangular algebras,unital prime algebras with a nontrivial idempotent,standard operator algebras and von Neumann algebras.We also prove that if T is a bounded linear operator from a unital C*-algebra A into a unital Banach algebra B such that the condition T(U)(?)T(V)=0 whenever UV=VU=0 and T(IA)=IB,then T is a Jordan homomorphism.In Chapter six,we give a summarization of the whole paper and pose some questions remaining unsolved.We also give some counter examples about the problems that we discuss in this paper,such as nontrivial inner derivations,nontrivial 2-local derivations and so on.