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.