LAUSR

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