On the Modular Theory of von Neumann Algebras

Abstract

The purpose of this thesis is to provide an exposition of the \textit{modular theory} of von Neumann algebras. The motivation of the theory is to classify and describe von Neumann algebras which do not admit a trace, and in particular, type III factors. We replace traces with weights, and for a von Neumann algebra $\mathcal{M}$ which admits a weight $\phi$, we show the existence of an automorphic action $\sigma^\phi:\mathbb{R}\rightarrow\text{Aut}(\mathcal{M})$. After showing the existence of these actions we can discuss the crossed product construction, which will then allow us to study the structure of the algebra.