$C^*$-simplicity and representations of topological full groups of groupoids

TL;DR

This paper investigates the conditions under which the topological full group of an ample groupoid generates its associated $C^*$-algebra, linking $C^*$-simplicity to the absence of tracial states and applying to groups acting on the Cantor set.

Contribution

It establishes a characterization of when the full group generates the $C^*$-algebra and provides new criteria for $C^*$-simplicity of topological full groups.

Findings

The image of the canonical representation generates $C^*(G)$ iff $C^*(G)$ has no tracial state.

Provides sufficient conditions for $C^*$-simplicity of certain topological full groups.

Applies results to groups acting freely and minimally on the Cantor set.

Abstract

Given an ample groupoid $G$ with compact unit space, we study the canonical representation of the topological full group $[[G]]$ in the full groupoid $C^*$-algebra $C^*(G)$. In particular, we show that the image of this representation generates $C^*(G)$ if and only if $C^*(G)$ admits no tracial state. The techniques that we use include the notion of groups covering groupoids. As an application, we provide sufficient conditions for $C^*$-simplicity of certain topological full groups, including those associated with topologically free and minimal actions of non-amenable and countable groups on the Cantor set.