Dense subalgebras of purely infinite simple groupoid C*-algebras

TL;DR

This paper characterizes when certain algebraic structures associated with groupoids are purely infinite, introducing an intermediate algebra to bridge algebraic and C*-algebraic properties, with applications to higher-rank graph algebras.

Contribution

It introduces the algebra $B(G)$ to connect algebraic pure infiniteness with C*-algebraic pure infiniteness for minimal, effective groupoids.

Findings

Algebraically purely infinite Steinberg algebras imply purely infinite simple reduced groupoid C*-algebras.

The algebra $B(G)$ is algebraically properly infinite if and only if $C^*_r(G)$ is purely infinite simple.

Results are applied to higher-rank graph 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 $C^*$-algebra $C^*_r(G)$ is simple and purely infinite. But the Steinberg algebra seems to small for the converse to hold. For this purpose we introduce an intermediate $*$-algebra $B(G)$ constructed using corners $1_U C^*_r(G) 1_U$ 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 $C^*_r(G)$ is purely infinite simple. We apply our results to the algebras of higher-rank graphs.