Simplicity of crossed products of the actions of totally disconnected locally compact groups on their boundaries

TL;DR

This work addresses when reduced crossed products arising from totally disconnected locally compact groups acting on their Furstenberg boundary are simple. It develops an intersection-property framework showing that topologically free boundary actions force the simplicity of C(∂F G) ⋊_r G, and establishes a partial converse under exactness. The approach extends classical discrete-group results (e.g., Kalantar–Kennedy, Kawamura–Tomiyama, Archbold–Spielberg) to the TDLC setting and applies to Neretin groups and HNN extensions, illustrating new non-discrete examples of simple crossed products. The findings contribute to understanding C*-simplicity for non-discrete groups and connect boundary dynamics with operator-algebraic simplicity, with potential implications for analyzing exactness and amenable actions in this broader context.

Abstract

We prove that if a totally disconnected locally compact group admits a topologically free boundary, then the reduced crossed product of continuous functions on its Furstenberg boundary by the group is simple. We also prove a partial converse of this result.

Theorems & Definitions (25)

• Theorem : See KalantarKennedy and BreuillardKalantarKennedyOzawa
• Theorem A: See Corollary \ref{['maincor']}
• Theorem B: See Corollary \ref{['conversecor']}
• Theorem 2.1
• Lemma 2.2
• Example 2.3
• Theorem 3.1
• Proposition 3.2
• proof
• Lemma 3.3