Research On Lie Mappings And Jordan Mappings Of Operator Algebras

In this dissertation, we study certain types of Lie mappings and Jordan map-pings on operator algebras. The research of this thesis focuses on nonlinear Lie derivation of triangular algebras, nonlinear *-Lie derivations on factor von Neumann algebras, nonlinear maps preserving *-Lie products and preservingξ-*-Lie products on factor von Neumann algebras, Jordan (θ,φ)-derivation of triangular algebras and generalized Jordan derivation of full matrix algebras. This dissertation is divided into four chapters.The first chapter is an introduction on the signigication and background of this thesis selecting subject. In addition, we offer necesary conceptions and conclusions for later chapters. Such conceptions and conclusions are with regard to derivation, inner derivation, triangular algebras, nest algebras and von Neumann algebras.In Chapter 2, firstly, we prove that every nonlinear Lie derivation of triangular algebras is the sum of an additive derivation and a map into its center sending commutators to zero. As an application, we describe the form of nonlinear Lie derivation on nest algebras and block upper triangular matrix algebras.In Chapter 3, to begin with we study nonlinear *-Lie derivation from a factor von Neumann algebra into itself and obtain that every nonlinear *-Lie derivation of factor von Neumann algebra is an additive *-derivation. Next we consider nonlinear bijective map preserving *-Lie product on factor von Neumann algebra so that it is a linear or conjugate linear *-isomorphism. Last but not least we show that every nonlinear bijective map preservingξ-*-Lie product on factor von Neumann algebra is additive.In the final Chapter, firstly, we are concerned with the Lie triple derivation from Alg(?) into anyσ-weakly closed algebra M. which contains Alg(?). We deduce that every Lie triple derivation is of the form X→XT - TX + h(X)I, where T∈M and h is a linear mapping from Alg(?) into C such that h([[A, B],C])= 0 for all A, B,C∈Alg(?). Secondly, we consider Jordan (θ,φ)-derivation of triangular algebras to be a (θ,φ)-derivation when triangular algebras Tri(A, M, B) only contain trivial idempotents. Finally, we prove that every generalized Jordan derivation of full matrix algebras is a sum of a derivation and a generalized inner derivation.The results from the thesis consist of the following statements.(1) Let U= Tri(A,M,B) be a triangular algebra and letφ:u→u be a nonlinear Lie derivation. IfπA(Z(u))= Z(A) andπB(Z(u))= Z(B). thenφis the sum of an additive derivation and a map into its center Z(u) sending each commutator to zero.(2) Let M be an factor von Neumann algebra acting on a complex Hilbert space H with dim H≥2. Suppose thatφ.M→M is a nonlinear *-Lie derivation. Thenφis an additive *-derivation.(3) Let H be a complex Hilbert space with dim H≥2 and let M. N be two factor von Neumann algebras acting on H. Suppose thatφ:M→N is a bijective map satisfyingφ(AB-BA*)=φ(4)φ(B)-φ(B)φ(4)* for all A. B∈M. Thenφis a linear or conjugate linear *-isomorphism.(4) Let H be a complex Hilbert space with dim H≥2 and let M.N be two factor von Neumann algebras acting on H. Letξ∈C andξ≠0.1. Suppose thatφ:M→N is a bijective map satisfyingφ(AB-BA*)=φ(A)φ(B)-ξφ(B)φ(A)* for all A. B∈M. Thenφis additive.(5) Let (?) be an independent finite-width CSL on a complex separable Hilbert space H. with dim H≥3, and M be anyσ-weakly closed algebra which contains Alg(?). If L:Alg(?)→M is a Lie triple derivation, then there exist T∈M and a linear mapping h from Alg(?) into C with h([[A, B].C])= 0 for all A, B, C∈Alg(?) such that L(X)= XT-TX+h(X)I for all X∈Alg(?).(6) Suppose thatθ,φare automorphisms of triangular algebras U= Tri(A,M, B). We prove that every Jordan (θ,φ)-derivation of triangular algebras is a (θ,φ)-derivation when triangular algebras Tri(A,M,B) only contain trivial idempotents.(7) Every generalized Jordan derivation of full matrix algebras is a sum of a derivation and a generalized inner derivation.