Abstract:
The aim of this thesis is to study closed two-sided ideals in a von Neumann algebra A, not only by looking into the structure of these ideals, but by using them in several applications on the theory of von Neumann algebras. For example, one of the main objects of this thesis is to develop a Riesz theory relative to any closed ideal in a von Neumann algebra by proving some characterization theorems of relatively Riesz operators and then to use this to prove a Riesz decomposition theorem. Section 1 contains the definitions of some basic facts concerning von Neumann algebras used throughout this work. The main issue of section 2 is to consider three specific examples of closed two-sided ideals in a semifinite algebra with a non-zero type I direct summand, namely the ideals of operators compact relative to the von Neumann algebra, the ideal of compact operators contained rn A and the ideal of the so called Rosenthal operators relative to A. These ideals are used to obtain factorization results as well as a duality theorem. In the third section we deduce geometrical characterizations as well as a spectral characterization for the quotient norm on A/1, where 1 is any closed ideal in A. We then prove some characterization theorems on the semi-Fredholm elements relative to 1. In section 4 Riesz operators relative to a closed two-sided ideal are defined. The results in this section are similar to those known for the classical case and they are used in the sequel to prove characterization theorems for relatively Riesz operators as well as a Riesz decomposition theorem. In section 5 a geometrical characterization of Riesz operators relative to any closed ideal is proved. This geometrical characterization is used in section 6 to obtain a Riesz decomposition theorem for Riesz operators relative to specific closed ideals in a semifinite van Neumann algebra.
