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

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 $G = \mathcal{F}(A)$.