We characterize properly purely infinite Steinberg algebras AK(G)\documentclass[12pt]{minimal}
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\begin{document}$$A_K({\mathcal {G}})$$\end{document} for strongly effective, ample Hausdorff groupoids G\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {G}}$$\end{document}. As an application, we show that the notions of pure infiniteness and proper pure infiniteness are equivalent for the Kumjian–Pask algebra KPK(Λ)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {KP}_K(\Lambda )$$\end{document} in case Λ\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda $$\end{document} is a strongly aperiodic k-graph. In particular, for unital cases, we give simple graph-theoretic criteria for the (proper) pure infiniteness of KPK(Λ)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {KP}_K(\Lambda )$$\end{document}. Furthermore, since the complex Steinberg algebra AC(G)\documentclass[12pt]{minimal}
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\begin{document}$$A_{\mathbb {C}}({\mathcal {G}})$$\end{document} is a dense subalgebra of the reduced groupoid C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^*$$\end{document}-algebra Cr∗(G)\documentclass[12pt]{minimal}
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\begin{document}$$C^*_r({\mathcal {G}})$$\end{document}, we focus on the problem that “when does the proper pure infiniteness of AC(G)\documentclass[12pt]{minimal}
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\begin{document}$$A_{\mathbb {C}}({\mathcal {G}})$$\end{document} imply that of Cr∗(G)\documentclass[12pt]{minimal}
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\begin{document}$$C^*_r({\mathcal {G}})$$\end{document} in the C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^*$$\end{document}-sense?”. In particular, we show that if the Kumjian–Pask algebra KPC(Λ)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {KP}_{\mathbb {C}}(\Lambda )$$\end{document} is purely infinite, then so is C∗(Λ)\documentclass[12pt]{minimal}
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\begin{document}$$C^*(\Lambda )$$\end{document} in the sense of Kirchberg–Rørdam.
