Fundamental group of finite von Neumann algebras with finite dimensional normal trace space

TL;DR

This paper introduces and characterizes the fundamental group of finite von Neumann algebras with finite dimensional trace space, showing its form is fully determined and can realize any group as such a fundamental group.

Contribution

It defines the fundamental group for these algebras and demonstrates that any group can be realized as the fundamental group of some such algebra.

Findings

The form of the fundamental group is completely determined.

Any group can be realized as the fundamental group of a finite von Neumann algebra with finite dimensional trace space.

Abstract

We introduce the fundamental group $F(\mathcal{M})$ of a finite von Neumann algebra $\mathcal{M}$ with finite dimensional normal trace space. The form of $F(\mathcal{M})$ is completely determined. Moreover, there exists a finite von Neumann algebra $\mathcal{M}$ with finite dimensional normal trace space such that $F(\mathcal{M})=G$ for any conceivable groups $G$ as fundamental group.