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

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 are equivalent Kumjian–Pask algebra \(\mathrm {KP}_K(\Lambda )\) in case \(\Lambda \) is a aperiodic k-graph. In particular, unital cases, give simple graph-theoretic criteria (proper) )\). Furthermore, since complex \(A_{\mathbb {C}}({\mathcal dense subalgebra reduced groupoid \(C^*\)-algebra \(C^*_r({\mathcal {G}})\), focus on problem “when does imply \(C^*\)-sense?”. if {KP}_{\mathbb {C}}(\Lambda infinite, then so \(C^*(\Lambda sense Kirchberg–Rørdam.