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Purely infinite $C^{\ast }$ -algebras associated to étale groupoids



Let $G$ be a Hausdorff, étale groupoid that is minimal and topologically principal. We show that $C_{r}^{\ast }(G)$ is purely infinite simple if and only if all the non-zero positive elements of $C_{0}(G^{(0)})$ are infinite in $C_{r}^{\ast }(G)$ . If $G$ is a Hausdorff, ample groupoid, then we show that $C_{r}^{\ast }(G)$ is purely infinite simple if and only if every non-zero projection in $C_{0}(G^{(0)})$ is infinite in $C_{r}^{\ast }(G)$ . We then show how this result applies to $k$ -graph $C^{\ast }$ -algebras. Finally, we investigate strongly purely infinite groupoid $C^{\ast }$ -algebras.



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Purely infinite $C^{\ast }$ -algebras associated to étale groupoids



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