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)

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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.
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
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

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
312369Groups, operator algebras, and dynamical systemsXin LiEuropean Commission (EC)101022531M&S - Mathematics