Wandering Vectors And Their Multipliers Of Finite Subdiagonal Algebras And Linear Mappings Preserving Rank-one

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.The theory of von Neumann algebra is a vast and very well-developrd area of the theory of operator algebras. The notion of subdiagonal algebras, a non-commutative analytic model of general von Neumann algebra in B(H),was introduced by Arveson to study the analyticity in operator algebras. Also Arveson introduced the notion of wandering vectors to study factorization in finite subdiagonal algebras. Wandering vectors play an important role in the analyticity of operator algebras, especially in studuing invariant subspaces of operator algebras.In this note, we consider wandering vectors of finite subdiagonal algebras,their multipliers and linear mappings preserving rank-one . The details as following :In chapter 1, some notations, definitions are introduced and some well-known theorems are given.In section I, we introduce the definitions of subdiagonal algebras, finite subdiagonal algebras,wandering vector,reflexive algebras, rank-one operator, operator toplogy and so on.In section II, some well-known theorems are given such as Kaplansky density theorem .In chapter 2, we consider wandering vectors and their multipliers for finite subdiagonal algebras. We first prove that the set of completely wandering vectors of a finite subdiagonal algebra is connected,second it is closed if and only if the finite subdiagonal algebra is antisymmetric.We last get the corollary that the set of completely wandering vectors of an antisymmetric finite subdiagonal algebra consistsof all unitary elements of M. In section II,we prove that the set of all wandering vector multipliers for an antisymmetric finite subdiagonal algebra forms a group.In chapter 3, we first recall the discussions of two invariant subspaces in reflexive algebras in history.Then using the ideas of lienar mappings preserving rank-one in nest algebra we discuss the lienar mappings preserving rank-one in the reflexive algebra about two invariant subspaces.