Local Jordan Mappings And Local Lie Mappings On Operator Algebras

In this dissertation,we mainly study certain types of local Jordan mappings and local Lie mappings on operator algebras.These mappings that we s-tudy include Jordan derivable mappings,Jordan higher derivable mappings,Jordan isomorphisms,local Lie derivations and Lie biderivations.This dissertation is divided into five chapters.In the first chapter,we introduce the significance and background of the select-ed topic,recall the present situation and achievements,and offer some necessary preliminary concepts and conclusions for later chapters.In Chapter 2,we first investigate Jordan derivable maps on triangular algebras and then two different characterizations of Jordan derivations on triangular algebras with some mild conditions are obtained.Next,we give a characterization of Jordan isomorphisms on triangular algebras.The results of this chapter are as follows.(1)Each Jordan derivable map on triangular algebras by commutative zero products is a generalized derivation.(2)Each Jordan derivable map at mutually inverse elements on triangular algebras is a derivation.(3)A linear bijection?:U?V is a Jordan isomorphism if and only if cp is unital and one of the following statements holds:(i)?(A ? B)= ?(A)? ?(B)for all A,B ?U with AB=0.(ii)?(A ? B)= ?(A)? ?(B)for all A,B ? U with A?B=0.(iii)?(A ? B)= ?(A)? ?(B)for all A,B ? U with AB=BA=0.In Chapter 3,we give two different characterizations of Jordan higher deriva-tions on triangular algebras through zero products and identity products.The results of this chapter are as follows.(4)Each Jordan higher derivable map on triangular algebras by commutative zero products is a generalized higher derivation.(5)Each Jordan higher derivable map at mutually inverse elements on trian-gular algebras is ahigher derivation.In Chapter 4,we consider local Lie derivations of operator algebras.Firstly,we show that under some mild conditions,each local Lie derivation of triangular algebras is a Lie derivation.Let ? be a non-trivial finite nest in a factor von Neumann algebra N.As an applications,we show that every local Lie derivation of AlgN? is a Lie derivation and each local Lie derivation of the join algebras is a Lie derivation.We also present an example to show that there exists a nontrivial local Lie derivation of triangular algebras.Secondly,we study local Lie derivations of factor von Neumann algebras.The results of this chapter are as follows.(6)Let U be-a triangular algebra.Suppose that(i)A and B are generated by their idempotents;(ii)?A(Z(U))= Z(A)and ?B(Z(U))= Z(B).Then each local Lie derivation of U is a Lie derivation.(7)Let A be a factor von Neumann algebra acting on a complex Hilbert space H with dim(A)? 2.Then each local Lie derivation ? from A into itself is a Lie derivation.In Chapter 5,we first study Lie biderivations of triangular algebras and give their structure.Also,a characterizations of a class of trilinear maps though zero Jordan triple product on the algebras generated by idempotents is obtained.The results of this chapter are as follows.(8)Each Lie biderivation is the sum of an extremal biderivation,a biderivation and a bilinear map into its center vanishing on all commutators in each argument.(9)Let A be an algebra generated by its idempotents.If ?:A×A×A?X is a trilinear map satisfying ?(A,B,C)= 0,whenever A?B?C=0.Then there exists a linear map f from A into X such that ?(A,B,C)= f(A ? B ?C)for any A,B,C?A.