Lie Isomorphisms And Lie Derivations Of Reflexive Algebras

This dissertation is devoted to the investigation of some Lie maps of reflexive al-gebras, including Lie isomorphisms of nest algebras on Banach spaces, local Lie deriva-tions and2-local Lie derivations on B(X), Lie triple derivations on JSL algebras and Lie triple derivable maps on B(X). It consists of five chapters.In the first chapter, we introduce the background, review the developments and achievements until now, and give preliminaries.In Chapter2, we first describe commutative Lie ideals in nest algebras and some related properties. Accordingly, a bijection of nontrivial subspaces of nests is induced. Furthermore, we prove that such a bijection is order-preserving or anti-order-preserving, and identify the behavior of the map on the idempotent operators and rank one opera-tors. Finally, we prove that a Lie isomorphism between nest algebras on Banach alge-bras can be written as a sum of an isomorphism or a negative of an anti-isomorphism, and a linear map with image in the center vanishing on each commutator. The result of this chapter is as follows.Theorem A. Let N and M be nests on Banach spaces X and Y, respectively. Suppose that ψ is a Lie isomorphism from the nest algebra AlgN onto the nest algebra AlgM Then one of the following holds.(i) There exists an invertible operator T in B(Y, X) and a linear functional τ on AlgN vanishing on each commutator such that ψ(A)=T-1AT+τ(A)I for all A∈AlgN.(ii) There exists an invertible operator T in B(Y,X*) and a linear functional τ on AlgN vanishing on each commutator such that ψ(A)=-T-1A*T+τ(A)I for all A∈AlgN.In Chapter3, we study the local Lie derivations and2-local Lie derivations on B(X). Making a use of known results of Lie derivations on B(X), we prove that every local Lie derivation on B(X) is a Lie derivations, and get a necessary and sufficient condition for the standardness of2-local Lie derivations on B(X). The results of this chapter are as follows.Theorem B. Suppose that X is a Banach space of dimension greater than2. Then every local Lie derivation from B(X) into itself is a Lie derivation. Theorem C. Let X be a Banach space of dimension greater than2. Then a map5:B(X)→B(X) is a2-local Lie derivation if and only if δ(A)=[A,T]+ψ(A) for all A∈B(X), where T∈B(X) and ψ is a homogeneous map from B(X) into FI satisfying ψ(A+B)=ψ(A) for A, B E B(X) with B being a sum of commutators.In Chapter4, we discuss Lie triple derivations, give the structure of Lie triple maps on JSL algebras and prove the weak additivity of Lie triple derivable maps on B(X). The results of this chapter are as follows.Theorem D. Let L be a J-subspace lattice on a Banach space X. Let δS:AlgL→AlgL be a linear map. The following are equivalent.(i) δ is a Lie triple derivation.(ii) For each K K∈J(L), there exists an operator TK in B(K) and a linear functional λK: AlgL→F vanishing on every double commutator such that δ(A)x=(TKA-ATK)x+λ(A)x, for all A∈A and x∈K.Theorem E. Let X be a Banach space of dimension greater than1. Suppose that a map δ:B(X)→B(X) satisfies δ([[A, B],C])=[[δ(A), B],C]+[[A,δ(B)], C]+[[A, B],δ(C)] for any A, B,C∈B(X). Then δ=D+τ, where D is an additive derivation of B(X) and the map τ:B(X)→FI satisfies τ([[A, B], C])=0for any A,B,C∈B(X).In Chapter5, we summarize the whole text, and put forward some questions remaining unsolved.