2-local Lie Derivations And 2-local Isomorphisms On Von Neumann Algebras

With the enrichment and development of the theory of derivations and isomorphisms,local Lie derivations?2-local Lie derivations and local isomorphisms?2-local isomorphisms have attracted the attention of many mathematicians.In this thesis,firstly we character-ize 2-local Lie derivations on matrix algebras Mn(C)and upper triangular matrix algebras Tn(C).Secondly,we investigate 2-local Lie derivations of algebras with unit.Finally,we investigate 2-local automorphism on Mn(A)(n?3),applying this result,we prove that a 2-local automorphism on an arbitrary AW*-algebra without type ?1 or ?2 direct summands is an automorphism.The structure of this thesis are as follows:In chapter two,we characterize 2-local Lie derivations on Mn(C)and Tn(C).Let Mn(C)be a matrix algebra,Tn(C)be an upper triangular matrix algebra.In this paper,we show that,if L:Mn(C)? Mn(C)is a 2-local Lie derivation,then there exist a matrix T ? Mn(C)and a map ?:Mn(C)?CIn such thatL(A)=TA-AT+?(A),(?)A ?Mn(C),where ?(A+F)=?(A),F=[A,B],VA,B E Mn(C).As its application,we show every 2-local Lie derivation from Mn1(C)(?)Mn2(C)(?)…(?)Mnm(C)into itself has the form L(A)=TA-AT+?(A).In addition,we show that,if L:Tn(C)?Tn(C)is a 2-local Lie derivation and satisfies L(A+B)-L(A)-L(B)?CIn,(?)A,B ?Tn(C),then L has the form L(A)=TA-AT+?(A).An example is given to show that the condition L(A+B)-L(A)-L(B)?CIn is necessary.2-Local Lie derivations from Tn1(C)(?)Tn2(C)(?)…(?)Tnm(C)into itself are also characterized.In chapter three,we investigate 2-local Lie derivations of algebras with unit.Let A be an unital algebra with the center Z(A),L:A?A be a 2-local Lie derivation.Under some mild assumptions on A and with the assumption L(A+B)-L(A)-L(B)E Z(A),(?)A,B ? A,we show that there exist an element T ? A and a homogeneous map f:A?Z(A)such that L(A)=TA-AT+f(A),(?)A?A.In particular,if Z(A)?{(?)[Ai,Bi},(?)Ai,Bi ? A,n ? N},then f(A+C)=f(A),C=[A,B],(?)A,B ? A.As its application,we characterize 2-local Lie derivations on B(X)and von Neumann algebra.In chapter four,we characterize surjective 2-local automorphisms on Mn(A)(n? 3).Let A be a Banach algebra,Mn(A)be the algebra of all n × n matrices over A.In this paper,we show that a surjective 2-local automorphism on Mn(A)(n? 3)is an automorphism.Applying this result,we prove that a surjective 2-local automorphism on an arbitrary AW*-algebra without type ?1 or ?2 direct summands is an automorphism.