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 tool in studying operator algebras. On the basis of existing papers, in this paper we mainly and detailedly discuss Jordan isomorphisms, local automorphisms, 2-local automorphisms, linear mappings preserving idempotents, linear mappings preserving zero product and linear mappings preserving Jordan zero product on nest subalgebras of factor von neumann algebras. 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 von Neumann algebras, factor von Neumann algebras, subspace lattices, nest algebras and bounded linear operators and so on. In section II, we give some well-known theorems, such as distinguished Erdos Jiechixing Theory.In chapter 2, we first discuss Jordan isomorphisms between two nest subalgebras of factor von Neumann algebras and prove that every Jordan isomorphism between two nest subalgebras of any factor von Neumann algebra is an isomorphism or an anti-isomorphism. Subsequently, we discuss local automorphisms and 2-local automorphisms on nest subalgebras of factor von Neumann algebras and prove that every linear surjective weakly continuous local automorphism is an automorphism and every linear surjective 2-local automorphism is an automorphism, respectively.