Algebraic actions I. C*-algebras and groupoids

Bruce, C. and Li, X. (2024) Algebraic actions I. C*-algebras and groupoids. Journal of Functional Analysis, 286(4), 110263. (doi: 10.1016/j.jfa.2023.110263)

Abstract

Given an algebraic action of a semigroup, we construct an inverse semigroup, and we characterize Hausdorffness, topological freeness, and minimality of the associated tight groupoid in terms of conditions on the initial algebraic action. We parameterize all closed invariant subspaces of the unit space of our groupoid, and characterize topological freeness of the associated reduction groupoids. We prove that our groupoids are purely infinite whenever they are minimal, which answers a general open question in the affirmative for our special class of groupoids. In the topologically free case, we prove that the concrete C*-algebra associated with the algebraic action is always a (possibly exotic) groupoid C*-algebra in the sense that it sits between the full and essential C*-algebras of our groupoid. This provides a framework for studying such concrete C*-algebras, allowing us to obtain structural results that were only previously available for very special classes of algebraic actions. For instance, we obtain results on simplicity and pure infiniteness for C*-algebras associated with subshifts over semigroups, actions coming from commutative algebra, and non-commutative rings. These results were out of reach using existing techniques.

• Item Type: Articles
• Additional Information: C. Bruce was supported by a Banting Fellowship administered by the Natural Sciences and Engineering Research Council of Canada (NSERC) and has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 101022531. X. Li has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 817597).
• Keywords: C*-algebras, etale groupoids, algebraic actions, inverse semigroups.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Bruce, Dr Chris and Li, Professor Xin
• Authors: Bruce, C., and Li, X.
• College/School: College of Science and Engineering > School of Mathematics and Statistics; College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Functional Analysis
• Publisher: Elsevier
• ISSN: 0022-1236
• ISSN (Online): 1096-0783
• Published Online: 29 November 2023
• Copyright Holders: Copyright © 2023 The Authors
• First Published: First published in Journal of Functional Analysis 286(4): 110263
• Publisher Policy: Reproduced under a Creative Commons License

Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
312369		Groups, operator algebras, and dynamical systems	Xin Li	European Commission (EC)	101022531	M&S - Mathematics