Fundamental group of simple $C^*$-algebras with unique trace II

TL;DR

This paper demonstrates that any countable subgroup of positive real numbers can be realized as the fundamental group of a separable simple unital $C^*$-algebra with a unique trace, and constructs many nonisomorphic examples for each subgroup.

Contribution

It shows the realization of arbitrary countable subgroups of positive reals as fundamental groups of simple $C^*$-algebras with unique trace, expanding the understanding of their possible fundamental groups.

Findings

Any countable subgroup of $ plus$ can be realized as a fundamental group.

For each such subgroup, there are uncountably many nonisomorphic algebras.

Constructs explicit examples of these algebras.

Abstract

We show that any countable subgroup of the multiplicative group $\mathbb{R}_+^{\times}$ of positive real numbers can be realized as the fundamental group $\mathcal{F}(A)$ of a separable simple unital $C^*$-algebra $A$ with unique trace. Furthermore for any fixed countable subgroup $G$ of $\mathbb{R}_+^{\times}$, there exist uncountably many mutually nonisomorphic such algebras $A$ with $G = \mathcal{F}(A)$.