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 $C^*$-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-R{\o}rdam.