RFD property for groupoid C*-algebras of amenable groupoids and for crossed products by amenable actions

[2305.12122] RFD property for groupoid C*-algebras of amenable groupoids and for crossed products by amenable actions Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Operator Algebras arXiv:2305.12122 (math) [Submitted on 20 May 2023 ( v1 ), last revised 11 Apr 2026 (this version, v3)] Title: RFD property for groupoid C*-algebras of amenable groupoids and for crossed products by amenable actions Authors: Tatiana Shulman , Adam Skalski View a PDF of the paper titled RFD property for groupoid C*-algebras of amenable groupoids and for crossed products by amenable actions, by Tatiana Shulman and 1 other authors View PDF HTML (experimental) Abstract: By Bekka's theorem the group C*-algebra of an amenable group $G$ is residually finite dimensional (RFD) if and only if $G$ is maximally almost periodic (MAP). We generalize this result in two directions of dynamical flavour. Firstly, we completely characterize the RFD property for crossed products by amenable actions of discrete groups on C*-algebras in terms of the action. The characterisation can be formulated in various terms, such as primitive ideals, (pure) states and approximations of representations, and the latter can be viewed as a dynamical version of Exel-Loring characterization of RFD C*-algebras. %The result leads among other consequences to a characterization of when a semidirect product by an amenable group has RFD full C*-algebra. As byproduct of our methods we characterize the property FD of Lubotzky and Shalom for semidirect products by amenable groups and obtain characterizations of the properties MAP and RF for general semidirect products of groups. These descriptions allow us to obtain the properties MAP, RF, RFD and FD for various new examples and generalize some results of Lubotzky and Shalom. Secondly, as another generalization of Bekka's theorem, we provide a sufficient condition and a necessary condition for the C*-algebra of an amenable étale groupoid to be RFD. Comments: 41 pages, v3 modifies an abstract, expands some proofs in Section 2, and adds further new examples in Section 5. The final version of the paper will appear in the Journal of Functional Analysis Subjects: Operator Algebras (math.OA) ; Group Theory (math.GR) MSC classes: Primary 46L05, Secondary 20E26, 46L55 Cite as: arXiv:2305.12122 [math.OA] (or arXiv:2305.12122v3 [math.OA] for this version) https://doi.org/10.48550/arXiv.2305.12122 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Adam Skalski [ view email ] [v1] Sat, 20 May 2023 06:59:29 UTC (41 KB) [v2] Tue, 19 Mar 2024 13:32:09 UTC (44 KB) [v3] Sat, 11 Apr 2026 15:20:52 UTC (45 KB) Full-text links: Access Paper: View a PDF of the paper titled RFD property for groupoid C*-algebras of amenable groupoids and for crossed products by amenable actions, by Tatiana Shulman and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.OA < prev | next > new | recent | 2023-05 Change to browse by: math math.GR References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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