Preserving Maps On Nest Subalgebras Of Von Neumann Algebras

The study of operator algebra bagan in 30times of 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 mordan 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 aglebra, such as linear maps that preserving zero Jordan triple products, linear maps preserving idempotent, and Jordan triple derivations at zero point. The datails as follows.In Chapter 1, we introduce some notions, definitions and some well-known theorems. We introduce some concepts, such as yon Neumann algebras, 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 zero Jordan triple product on nest subalgebrasof factor yon Neumann algebras. We prove that every linear map preserving zero Jordan triple product and unit from nest subalgebras of factor yon Neumann algebras to Banach algebra with unit is a Jordan isomorphisms. We also discuss linear maps preserving zero product on certain reflexive operator algebras whose lattices cointain a non-trivial comparable element and show that such preserving maps are isomorphisms.In Chapter 3, we pay our attention on Jordan triple derivable maps at zero point of nest subalgebras of factor von Neumann algebras. It is proved that such a map is the sum of a derivation and the identity map.In Chapter 4, we discuss the linear map that preserving idempotent between two nest subalgebras of factor von Neumann algebras, and prove that every linear surjective map preserving idempotent between two nest subalgebras of factor von Neumann algebras is an isomorphism or an anti-isomorphism.