C⁎-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⁎-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.