Dense subalgebras of purely infinite simple groupoidC*-algebras

Abstract

AbstractA simple Steinberg algebra associated to an ample Hausdorff groupoidGis 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 too small for the converse to hold. For this purpose we introduce an intermediate *-algebraB(G) constructed using corners$1_U C^*_r(G) 1_U$for all compact open subsetsUof the unit space of the groupoid. We then show that ifGis minimal and effective, thenB(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.