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

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.