C*-simplicity of locally compact Powers groups

TL;DR

This paper explores the simplicity of C*-algebras associated with locally compact groups acting on trees, establishing foundational results and providing the first simplicity proof for non-discrete groups.

Contribution

It introduces the concept of C*-simplicity for locally compact groups, proves simplicity for certain non-discrete groups, and extends Powers' property to this setting.

Findings

Every C*-simple group is totally disconnected.

Reduced group C*-algebras of certain groups acting on trees are simple.

Group von Neumann algebras of these groups are factorial, non-amenable, and of specific types.

Abstract

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.