Dense subalgebras of purely infinite simple groupoid C*-algebras

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 -algebra is simple and purely infinite. But the Steinberg algebra seems too small for the converse to hold. For this purpose we introduce an intermediate ∗-algebra B(G) constructed using corners 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 is purely infinite simple. We apply our results to the algebras of higher-rank graphs.