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 Ⅰ1 or Ⅰ2 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 τ:Mn(C)→CIn such thatL(A)=TA-AT+τ(A),(?)A ∈Mn(C),where τ(A+F)=τ(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+τ(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+τ(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 Ⅰ1 or Ⅰ2 direct summands is an automorphism.