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

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