Modular Theory by example

TL;DR

This paper introduces Modular Theory for von Neumann algebras, explaining key results like the Tomita-Takesaki theorem, and illustrates these concepts with elementary and advanced examples including crossed products and CAR-algebras.

Contribution

It provides a clear introduction to Modular Theory with detailed examples, including computations for complex algebraic structures, enhancing understanding of the theory's applications.

Findings

Formulation of the Tomita-Takesaki theorem

Examples of modular objects in crossed products

Calculations for CAR-algebra in Fock representation

Abstract

The present article contains a short introduction to Modular Theory for von Neumann algebras with a cyclic and separating vector. It includes the formulation of the central result in this area, the Tomita-Takesaki theorem, and several of its consequences. We illustrate this theory through several elementary examples. We also present more elaborate examples and compute modular objects for a discrete crossed product and for the algebra of canonical anticommutation relations (CAR-algebra) in a Fock representation.