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.
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Accepted:
Published online:
Keywords: Uniform Roe algebras, coarse geometry, Cartan subalgebra
Mot clés : Algèbres de Roe uniformes, géométrie grossière, sous-algèbres de Cartan
Braga, Bruno M. 1; Farah, Ilijas 2, 3; Vignati, Alessandro 4
@article{AIF_2022__72_1_301_0, author = {Braga, Bruno M. and Farah, Ilijas and Vignati, Alessandro}, title = {General {Uniform} {Roe} algebra rigidity}, journal = {Annales de l'Institut Fourier}, pages = {301--337}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {72}, number = {1}, year = {2022}, doi = {10.5802/aif.3461}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3461/} }
TY - JOUR AU - Braga, Bruno M. AU - Farah, Ilijas AU - Vignati, Alessandro TI - General Uniform Roe algebra rigidity JO - Annales de l'Institut Fourier PY - 2022 SP - 301 EP - 337 VL - 72 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3461/ DO - 10.5802/aif.3461 LA - en ID - AIF_2022__72_1_301_0 ER -
%0 Journal Article %A Braga, Bruno M. %A Farah, Ilijas %A Vignati, Alessandro %T General Uniform Roe algebra rigidity %J Annales de l'Institut Fourier %D 2022 %P 301-337 %V 72 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3461/ %R 10.5802/aif.3461 %G en %F AIF_2022__72_1_301_0
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. https://aif.centre-mersenne.org/articles/10.5802/aif.3461/
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