Dense subalgebras of purely infinite simple groupoid C*-algebras

Abstract

A simple Steinberg algebra associated to an ample Hausdorff groupoid $G$ is algebraically purely infinite if and only if the characteristic functions of compact open subsets of the unit space are infinite idempotents. If a simple Steinberg algebra is algebraically purely infinite, then the reduced groupoid $C^{\ast }$-algebra $C_r^{\ast }(G)$ is simple and purely infinite. But the Steinberg algebra seems to small for the converse to hold. For this purpose we introduce an intermediate $\ast$-algebra $B(G)$ constructed using corners $1_U{C}_r^{\ast }(G)1_U$ for all compact open subsets $U$ of the unit space of the groupoid. We then show that if $G$ is minimal and effective, then $B(G)$ is algebraically properly infinite if and only if $C_r^{\ast }(G)$ is purely infinite simple. We apply our results to the algebras of higher-rank graphs.

To appear in Proceedings of the Edinburgh Mathematical Society. Minor corrections and updated references

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• Higher rank graph C*-algebras
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• Purely infinite $C^{\ast }$-algebras associated to étale groupoids
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• Graph-Based Models for Kirchberg Algebras