Non-simple purely infinite Steinberg Algebras with applications to Kumjian-Pask algebras

Abstract

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