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

• Autores: Hossein Larki
• Localización: Mediterranean journal of mathematics, ISSN 1660-5446, Vol. 18, Nº. 2, 2021
• Idioma: inglés
• DOI: 10.1007/s00009-020-01695-0
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• Resumen

  • We characterize properly purely infinite Steinberg algebras AK(G) for strongly effective, ample Hausdorff groupoids G. As an application, we show that the notions of pure infiniteness and proper pure infiniteness are equivalent for the Kumjian–Pask algebra KPK(Λ) in case Λ 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). Furthermore, since the complex Steinberg algebra AC(G) is a dense subalgebra of the reduced groupoid C∗-algebra C∗r(G), we focus on the problem that “when does the proper pure infiniteness of AC(G) imply that of C∗r(G) in the C∗-sense?”. In particular, we show that if the Kumjian–Pask algebra KPC(Λ) is purely infinite, then so is C∗(Λ) in the sense of Kirchberg–Rørdam.