  In this article we initiate research on locally compact C*-simple groups. We
first show that every C*-simple group must be totally disconnected. Then we
study C*-algebras and von Neumann algebras associated with certain groups
acting on trees. After formulating a locally compact analogue of Powers'
property, we prove that the reduced group C*-algebra of such groups is simple.
This is the first simplicity result for C*-algebras of non-discrete groups and
answers a question of de la Harpe. We also consider group von Neumann algebras
of certain non-discrete groups acting on trees. We prove factoriality,
determine their type and show non-amenability. We end the article by giving
natural examples of groups satisfying the hypotheses of our work.
