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Purely infinite $C^*$-algebras associated to étale groupoids (1403.4959v1)
Published 19 Mar 2014 in math.OA
Abstract: Let $G$ be a Hausdorff, \'etale groupoid that is minimal and topologically principal. We show that $C*_r(G)$ is purely infinite simple if and only if all the nonzero positive elements of $C_0(G0)$ are infinite in $C_r*(G)$. If $G$ is a Hausdorff, ample groupoid, then we show that $C*_r(G)$ is purely infinite simple if and only if every nonzero projection in $C_0(G0)$ is infinite in $C*_r(G)$. We then show how this result applies to $k$-graph $C*$-algebras. Finally, we investigate strongly purely infinite groupoid $C*$-algebras.