Lie On Operator Algebra - Factor Guide, The Left Guide And 2 - Local Guide

Operator algebra plays an important role in modern mathematics. During these years, many mathematicians have done good work in this field. Inspired by some known results on Lie derivation, this thesis aims at studying some mappings on operator algebras which include derivations, Lie-derivations, Jordan-derivations, Lie-ξ derivations, left derivations, Jordan left derivations and2-local derivations. The operator algebras include generalized matrix algebras, Banach algebras,von Neumann and J-subspace lattice algebras.The thesis is divided into five chapters. The first chapter introduces some background knowledge of operator algebras and presents a summary of some good results on operator algebras. Also, we give the readers some concepts which will be used in this thesis.In chapter2,we study Lie-ξ derivations on generalized matrix algebras. Let A be a unital algebra over a unital ring R.A linear mapping LeL(A) is called a Lie-ξ derivation if it satisfies L(ab-ξba)=L(a)b+aL(b)-ξL(b)a-ξbL(a) for every a,beA. It is shown that every Lie-ξ derivation on A derivable on0is a derivation whenξ is not1or-1.And in the remaining part, we discuss the Lie-ξ derivation derivable on IA(?)0and I with similar methods.In chapter3, some properties of left derivations were studied. It is shown that if δ is an additive mapping from a2-torsion free semi-prime ring R into M, and satisfies δ(xyx)=2xyδ(x)+x2δ(y) and xyδ(x)=yxδ(x) for every x,y∈R, then8is a derivation. Also, it is shown that5is left derivable at I if and only if δ is a Jordan left derivation.In chapter4, we give some results of2-local derivations on J-subspace lattice algebra.In the last chapter, we give a summarization of this thesis, and pose some unsolved questions.