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

Abstract

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 nonzero 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 nonzero 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.

References in corpus (7)

• Higher rank graph C*-algebras
• Simplicity of C*-algebras associated to higher-rank graphs
• Purely infinite C*-algebras arising from crossed products
• A groupoid generalization of Leavitt path algebras
• Strong pure infiniteness of crossed products
• Gauge-invariant ideals in the C*-algebras of finitely aligned higher-rank graphs
• The structure of a the C*-algebra of a locally injective surjection

Cited by in corpus (7)

• Ideal structure and pure infiniteness of ample groupoid $C^{\ast }$-algebras
• $K$-theory for real $k$-graph $C^{\ast }$-algebras
• A dichotomy for simple self-similar graph $C^{\ast }$-algebras
• Real rank and topological dimension of higher rank graph algebras
• Pure infiniteness and paradoxicality for graph $C^{\ast }$-algebras
• Dense subalgebras of purely infinite simple groupoid C*-algebras
• AF-embeddability of $2$-graph algebras and quasidiagonality of $k$-graph algebras