Non-simple Purely Infinite Steinberg Algebras with Applications to Kumjian–Pask Algebras

Abstract

We characterize properly purely infinite Steinberg algebras $A_K({\mathcal {G}})$ for strongly effective, ample Hausdorff groupoids ${\mathcal {G}}$. As an application, we show that the notions of pure infiniteness and proper pure infiniteness are equivalent for the Kumjian–Pask algebra $\mathrm {KP}_K(\Lambda )$ in case $\Lambda$ is a strongly aperiodic k-graph. In particular, for unital cases, we give simple graph-theoretic criteria for the (proper) pure infiniteness of $\mathrm {KP}_K(\Lambda )$. Furthermore, since the complex Steinberg algebra $A_{\mathbb {C}}({\mathcal {G}})$ is a dense subalgebra of the reduced groupoid $C^*$-algebra $C^*_r({\mathcal {G}})$, we focus on the problem that “when does the proper pure infiniteness of $A_{\mathbb {C}}({\mathcal {G}})$ imply that of $C^*_r({\mathcal {G}})$ in the $C^*$-sense?”. In particular, we show that if the Kumjian–Pask algebra $\mathrm {KP}_{\mathbb {C}}(\Lambda )$ is purely infinite, then so is $C^*(\Lambda )$ in the sense of Kirchberg–Rørdam.