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arXiv · 1805.06743

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

Kevin Aguyar Brix·Eduardo Scarparo
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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.

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Kevin Aguyar Brix, Eduardo Scarparo. 2018-05-17. $C^*$-simplicity and representations of topological full groups of groupoids. https://doi.org/10.1016/j.jfa.2019.06.014

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