  Suzuki recently gave constructions of non-discrete examples of locally
compact C*-simple groups and Raum showed C*-simplicity of the relative
profinite completions of the Baumslag-Solitar groups by using Suzuki's results.
We extend this result to some fundamental groups of graphs of groups called
generalized Baumslag-Solitar groups. In this article, we focus on some
sufficient condition to show that these locally compact groups are C*-simple
and that KMS-weights of these reduced group C*-algebras are unique. This
condition is an analogue of the Powers averaging property of discrete groups
and holds for several currently known constructions of non-discrete C*-simple
groups.
