  We introduce the fundamental group F(A) of a simple $\sigma$-unital
$C^*$-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 $C^*$-algebras.
We show that there exist separable stably projectionless $C^*$-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.
