Linear Mappings On Nest Subalgebras Of Factor Von Neumann Algebras

The study of operator algebra theory began in 30times of the 20th century. With the fast development of the theory, now it has become a hot branch playing the role of an initiator in morden mathematics. It has unexpected relations and interinfiltrations with quantum mechanics, noncommutative geometry, linear system and control theory, indeed number theory as well as some other important branches of mathematics. In order to discuss the structure of operator algebras, in recent years, many scholars both here and abroad have focused on linear mappings on operator algebras and have been introduced more and more new methods. For example, local mappings, 2-local mappings, dual local mappings, elementary mappings, linear preserving problems and so on were introduced successively, at present time these mappings have become important tools in studying operator algebras. Nest algebra is a class of most important non-semisimple and non-selfadjoint operator algebra. Its finite dimensional model is upper triangular matrix algebra, but the infinite dimensional model is more complex. On the basis of existing papers, in this paper we mainly and detailedly discuss local φ-derivations. 2-local φ-derivations. generalized φ-derivations and local 2-cocycles on nest subalgebras of factor von Neumann algebras. In addition, we study Jordan (α,α)-derivations on semiprime rings. The details as following:In chapter 1. some notations, definitions are introduced and some well-known theorems are given. In section I. we give some technologies and notations, and introduce the definitions of derivations, inner derivations, local derivations, generalized derivations, 2-local derivations, factor von Neumann algebras, nest algebras and so on. In section II, we give some well-known theorems, such as distinguished Erdos Density Theory.In chapter 2, we first discuss local φ-derivations on nest algebras and prove that every norm continuous local φ-derivation on nest algebras is a φ-derivation. Subsequently, we discuss 2-local φ-derivations on nest algebras and prove that every 2-local φ-derivation on nest algebras is a φ-derivation. Lastly, we discuss the generalized φ-derivations on nest algebras, detailedlv and obtain a list of theorems andcorollaries.In chapter 3, we first discuss local 2-cocycles on nest subalgebras of factor von Neumann algebras and prove that every weakly continuous local 2-cocycle is a 2-cocycle. Subsequently, we character and study the 2-cocycles on a subalgebra of the algebra M3(C) and obtain the necessary and sufficient conditions that a bilinear mapping is a 2-cocycle on this algebra.In chapter 4, we discuss generalized Jordan (a,a)-derivations on 2-torsion free semiprime rings and get that every generalized Jordan (<*, ev)-derivation on 2-torsion free semiprime rings is a generalized (a, a)-derivation.