A locally compact group is called C*-simple if its reduced group C*-algebra is simple. C*-simplicity of discrete groups is well-studied and characterized by boundary actions.
On the other hand, we don't have such results for non-discrete groups, and there are few examples. In this talk, I'll introduce a construction of non-discrete examples of C*-simple groups acting on trees. Recently, Raum showed C*-simplicity of the Shulitching completions of Baumslag-Solitar groups. I'll explain another proof of this result by using an analogue of the Powers averaging property.