Fundamental group of simple $C^{\ast }$-algebras with unique trace III

Abstract

We introduce the fundamental group F(A) of a simple $\sigma$-unital $C^{\ast }$-algebra $A$ with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of our previous works. Our definition in this paper makes sense for stably projectionless $C^{\ast }$-algebras. We show that there exist separable stably projectionless $C^{\ast }$-algebras such that their fundamental groups are equal to ${\mathbb{R}}_{+}^{\times }$ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case. This study is motivated by the work of Kishimoto and Kumjian.

15 pages, Typos are corrected