Types of von Neumann Algebras

Abstract

In this thesis we examine the classification theory regarding von Neumann algebras. We will focus on the classification of von Neumann algebras into types, namely type I, II and III. Furthermore the decomposition of von Neumann algebra into a direct integral of factors is included. The classification process starts with abelian von Neumann algebras. After the abelian von Neumann algebras are classified we examine the general structure of type I and type II von Neumann algebras. We then proceed to provide explicit constructions of factors of the possible types. The focus lies with the general structure theory of von Neumann algebras. Various constructions, such as, the crossed product construction are included, as it provides us with a tool to construct factors of type II1, type II1 and type III.