Local Maps Of Reflexive Operator Algebras

This dissertation is devoted to the investigation of some local maps of reflexive algebras, including 2-local Lie isomorphisms and multiplicative maps that close to an automorphism of B(X); local Lie derivations of nest algebras on Hilbert spaces; local Lie derivations,2-local derivations,2-local isomorphisms and derivable mappings of JSL algebras. It consists of six chapters.In the first chapter, we introduce the background, review the developments and achievements until now, and give preliminaries.In Chapter 2, we study the multiplicative maps that close to an automorphism and give a new characterization of automorphism on B(X). The result of this chapter is as follows.Theorem A Let X be a Banach space of dimension greater than one and 0< δ< 1/λ2(X)(if X is reflexive,0<δ< 1). Let φ:B(X)â†'B(X) be a linear automorphism and φ:B(X)â†'B(X) be a multiplicative map satisfying ‖φ(A)-φ(A)‖≤δ‖φ(A)‖ for all 0≠A∈B(X). Then there exists an invertible operator T ∈ B(X) such that φ(A)=T-1 AT, for all A e B(X). (λ2(X) is a projection constant).In Chapter 3, we study the surjective 2-local Lie isomorphisms from B(X) into B(Y). The result of this chapter is as follows.Theorem B Let X and Y be complex Banach spaces of dimension greater than 2. Let φ be a surjective 2-local Lie isomorphism from B(X) onto B(Y).Then one of the following holds.(1) φ=φ+Ï„, where φ is an isomorphism from B(X) onto B(Y), and Ï„ is a homo- geneous map from B(X) into CI vanishing on all finite sums of commutators.(2)Φ=-φ+Ï„, where φ is an anti-isomorphism from B(X) onto B(Y), and Ï„ is a homogeneous map from B(X) into CI vanishing on all finite sums of commu-tators.In Chapter 4, we study the local Lie derivations of nest algebras on Hilbert space. Making a use of known result of Lie derivation on nest algebra, we prove that every local Lie derivation on nest algebra is a Lie derivation. The result of this chapter is as follows.Theorem C Let N be a nontrivial nest on a Hilbert space H and AlgN be the associated nest algebra. Then every local Lie derivation δ from AlgN into B(H) is a Lie derivation.In Chapter 5, firstly, we discuss local Lie derivations on JSL algebra; secondly, we study 2-local derivations and 2-local isomorphisms on the standard subalgebras of JSL algebra; at last, we give a new characterization of generalized derivations on JSL algebra. The results of this chapter are as follows.Theorem D. Let L be a J-subspace lattice on a Banach space X. Then every local Lie derivation from AlgL into itself is a Lie derivation.Theorem E Let L be a J-subspace lattice on a Banach space X and A be a standard subalgebra of AlgL. If δ:Aâ†'B(X) is a 2-local derivation, then δ is a derivation.Theorem F Let Li be a J-subspace lattice on a Banach space Xi and Ai be a standard subalgebra of AlgLi, i= 1,2.If Φ is a surjective 2-local isomorphism from A1 onto A2, then Φ is an isomorphism.Theorem G Let L be a J-subspace lattice on a Banach space X and M be in AlgL. Suppose that δ:Alg Lâ†'B(X) is a linear mapping and derivable on the relation R={(A,B)∈AlgL×AlgL:AMB= 0}. Then δ is a generalized derivation. Moreover, for each K∈J(L), there holds δ(I)|K=λKM|K for some λK∈F.In Chapter 6, we summarize the whole text, and put forward some questions remaining unsolved.