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

Kaplansky introduced the notions of CCR and GCR C^{\ast }$C^{\ast }$-algebras, because they have a tractable representation theory. Many years later, he introduced the notions of CCR and GCR rings. In this paper we characterize when the algebra of an ample groupoid over a field is CCR and GCR. The results turn out to be exact analogues of the corresponding characterization of locally compact groupoids with CCR and GCR C^{\ast }$C^{\ast }$-algebras. As a consequence, we classify the CCR and GCR Leavitt path algebras.