Additivity Of Jordan Maps And Biderivations On Nest Subalgebras

The study of operator algebra began in 30times of the 20th century. Though compary with some other theory it is relatively new, but it has unexpected application in some mathematic theory and other subject, such as quantum mechanics, noncommutative geometry, linear system, contral theory, number theory and some other important branches of mathematics. Accompany with its using in other subjects, this theory developed a lot. Now it has become a hot branch in mor-dan mathematics. The class of non-selfadjiont operator algebras is an important domain in operator algebra reaserching. And nest algebras are the most important kind in non-selfadjiont operator algebras. In recent years, many scholar both here and abroad have focused on them a lot. They have done many works, not only raising many new thinkings, but also introducing many advanced methods. In this paper we pay our attention on some maps on nest algebras and nest subalgebras of factor von Neumann algebra, such as linear maps that preserving anti-zero product, maps that can additive automatically, biderivation and generalized biderivation. The datails as following.In chapter 1, we introduce some notions, definitions and some well-known theorems. We introduced some concopts, such as factor von Neumann algebras, nest algebras and so on, and give some well-known theorems that we will use in this paper.In chapter 2, we put our attention on linear maps that preserving anti-zero product on nest subalgebras of factor von Neumann algebras. First, we proved that every linear maps that preserving anti-zero product and unit from one nest subalgebra of factor von Neumann algebra to another is an anti-isomorphism. Then we proved that every linear maps that preserving anti-zero product but without unit on nest subalgebras of factor von Neumann algebras to itself is a non-zero scalar multiply anti-automorphism.In chapter 3, we discuss some maps that can additive automatically. First, we proved Jordan maps with some condition that on nest subalgebras of factor von Neumann albebras can additive automatically. Then we proved Jordan elementary maps with some condition that on nest subalgebras of factor von Neumann algebras can additive automatically also.In chapter 4, we pay our attention on biderivation and generalized biderivations of nest algebras. We give a characterisation such that every biderivation on it is an inner biderivation. Then we discuss generalized biderivation and prove that every generalized biderivation is an inner generalized biderivation under some conditions.