General Uniform Roe algebra rigidity
Annales de l'Institut Fourier, Volume 72 (2022) no. 1, pp. 301-337.

We generalize all known results on rigidity of uniform Roe algebras to the setting of arbitrary uniformly locally finite coarse spaces. For instance, we show that isomorphism between uniform Roe algebras of uniformly locally finite coarse spaces whose uniform Roe algebras contain only compact ghost projections implies that the base spaces are coarsely equivalent. Moreover, if one of the spaces has property A, then the base spaces are bijectively coarsely equivalent. We also provide a characterization for the existence of an embedding onto hereditary subalgebra in terms of the underlying spaces. As an application, we partially answer a question of White and Willett about Cartan subalgebras of uniform Roe algebras.

Nous généralisons tous les résultats connus sur la rigidité des algèbres de Roe uniformes au cas des espaces grossiers uniformément localement finis arbitraires. Nous montrons que l’isomorphisme entre les algèbres de Roe uniformes des espaces grossiers uniformément localement finis dont les algèbres de Roe uniformes ne contiennent que des projections fantômes compactes implique que les espaces de base sont grossièrement équivalents. De plus, si l’un des espaces a la propriété A, alors les espaces de base sont bijectivement grossièrement équivalents. Nous fournissons également une caractérisation de l’existence d’un plongement sur les sous-algèbres héréditaires en termes de sous-espaces. Comme application, nous répondons partiellement à une question de White et Willett sur les sous-algèbres de Cartan des algèbres de Roe uniformes.

Published online:
DOI: 10.5802/aif.3461
Classification: 46L05, 46L85, 51F30
Keywords: Uniform Roe algebras, coarse geometry, Cartan subalgebra
Braga, Bruno M. 1; Farah, Ilijas 2, 3; Vignati, Alessandro 4

1 University of Virginia 141 Cabell Drive Kerchof Hall, P.O. Box 400137, Charlottesville (USA)
2 Department of Mathematics and Statistics York University 4700 Keele Street Toronto, Ontario, M3J 1P3 (Canada)
3 Matematički Institut SANU Kneza Mihaila 36 11 000 Beograd, p.p. 367 (Serbia)
4 Université Paris Cité, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG) Bâtiment Sophie Germain 8 Place Aurélie Nemours 75013 Paris (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {General {Uniform} {Roe} algebra rigidity},
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Braga, Bruno M.; Farah, Ilijas; Vignati, Alessandro. General Uniform Roe algebra rigidity. Annales de l'Institut Fourier, Volume 72 (2022) no. 1, pp. 301-337. doi : 10.5802/aif.3461.

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