$C^{\ast }$-simplicity and representations of topological full groups of groupoids

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 $C^{\ast }$-algebra $C^{\ast }(G)$. In particular, we show that the image of this representation generates $C^{\ast }(G)$ if and only if $C^{\ast }(G)$ admits no tracial state. The techniques that we use include the notion of groups covering groupoids. As an application, we provide sufficient conditions for $C^{\ast }$-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.

arXiv comment13 pages. Minor changes, including new references. Accepted in Journal of Functional Analysis

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Cited by in corpus (3)

• The Furstenberg Boundary of a Groupoid
• The ideal intersection property for essential groupoid C*-algebras
• Representing topological full groups in Steinberg algebras and C*-algebras