Fundamental group of $AF$-algebras with finite dimensional trace space

TL;DR

This paper investigates the fundamental groups of AF-algebras with finite-dimensional trace spaces, revealing cases not realizable as fundamental groups of von Neumann algebras, thus expanding understanding of their algebraic structure.

Contribution

It identifies fundamental groups of AF-algebras with finite-dimensional trace spaces that cannot be realized as von Neumann algebra fundamental groups, highlighting new distinctions.

Findings

Certain AF-algebras have fundamental groups not realizable as von Neumann algebra groups

The paper characterizes the fundamental groups of AF-algebras with finite-dimensional trace spaces

New examples of algebraic structures with unique fundamental groups are provided

Abstract

We consider the realization of fundamental groups of $AF$-algebras in a certain class. We find the fundametal groups of $AF$-algebras with finite dimensional trace space which is not realizable as a fundamental group of von Neumann algebras.