Research Of Some Derivable Mappings On Operator Algebras

Abstract In this dissertation, we pay our attention to some derivable mappings and products preserving mappings on operator algebras. The mappings we discuss include κ-Jordan derivable mappings, κ-Jordan triple derivable mappings, Jordan*—derivable mappings, Lie triple derivable mappings, ξ-Lie derivable mappings and Lie higher derivable mappings, and so on. The operator algebras we discuss include standard operator algebras, von Neumann algebras, nest algebras and triangular alge-bras.This dissertation is divided into five chapters. The contents are as follows.In chapter1, we introduce the signification of the thesis selecting subject and recall some result about the subject. In addition, we offer some necessary knowledge about operator algebras that we need in this dissertation.In chapter2, we introduce the concept of κ-ordan derivable mapping. Let A be an algebra over C, and κ:is a fixed nonzero rational number. A map δ:A→A is called a κ-Jordan derivable mapping if δ(k(ab+ba))=k[δ(a)b+ad(b)+δ(b)a+bδ(a)],(?)a. b∈A. We prove that every κ-Jordan derivable mapping of nest algebras is an additive derivation. At the same time, we describe the forms of κ-Jordan derivable mappings of nest algebras acting on an infinite-dimensional Hilbert space. Related results concerning κ-ordan triple derivable maps are given. Finally, we prove that every Lie triple derivable map of nest algebras is the sum of an additive derivation and a map from algβ into C which maps second commutator into zero. We also describe the forms of Lie triple derivable mappings of nest algebras acting on an infinite-dimensional Hilbert space.In chapter3, we discuss the ξ-Lie derivable mappings from the standard operator algebra of B(X) to the discussed B(X). It is shown that such a mapping with ξ=1is the sum of an additive derivation and a map from the standard operator algebra into C which maps commutators into zero; the mapping with ξ≠1is an additive derivation satisfying δ(ξx)=ξδ(x),(?)x∈A. Related results concerning generalized ξ-Lie derivable maps are also given. Next, we study the Jordan*-derivable mapping δ from the standard operator algebra of B(H) to the discussed B(H), that is, δ(ab+ba)=δ(a)b*+aδ(b)+δ(b)a*+bδ(a).(?)a, b∈A. We deduce that δ is the form of δ(a)=aT-Ta*, where T∈B(H), for all a∈A.In chapter4, we first show the mapping of triangular algebras that Jordan deriv-able at0is an additive derivation, and deduce the mapping that ξ-Lie derivable at standard idempotent is a derivation. Secondly, we prove the Jordan higher derivable mapping at0is an additive higher derivation, and the ξ-Lie higher derivable mapping at standard idempotent is a higher derivation. Finally, we prove the nonlinear Lie higher derivable mapping of triangular algebra is the sum of a higher derivation and a mapping from the triangular algebra into its center which maps commutators into zero.In chapter5, we introduce the concept of map preserving the Jordan*-product, that is, the bijective mapΦ:A→Β satisfiesΦ(X*Y+YX*)=Φ(X)*Φ(Y)+Φ(Y)Φ(X)*,(?)X, Y∈A.We show the map preserving the Jordan*-product on factor von Neumann algebras is a*-ing isomorphism. We also describe the forms of Φ acting on typeⅠfactors or finite factors.