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Transactions of the American Mathematical Society

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On the rigidity of uniform Roe algebras over uniformly locally finite coarse spaces


Authors: B. M. Braga and I. Farah
Journal: Trans. Amer. Math. Soc. 374 (2021), 1007-1040
MSC (2010): Primary 46L80, 46L85, 51K05
DOI: https://doi.org/10.1090/tran/8180
Published electronically: November 3, 2020
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Abstract: Given a coarse space $ (X,\mathcal {E})$, one can define a $ \mathrm {C}^{*}$-algebra $ \mathrm {C}^{*}_{u}(X)$ called the uniform Roe algebra of $ (X,\mathcal {E})$. It has been proved by J. Špakula and R. Willett that if the uniform Roe algebras of two uniformly locally finite metric spaces with property A are isomorphic, then the metric spaces are coarsely equivalent to each other. In this paper, we look at the problem of generalizing this result for general coarse spaces and on weakening the hypothesis of the spaces having property A.


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Additional Information

B. M. Braga
Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, M3J IP3, Canada
Email: demendoncabraga@gmail.com

I. Farah
Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, M3J IP3, Canada
Email: ifarah@mathstat.yorku.ca

DOI: https://doi.org/10.1090/tran/8180
Keywords: Bounded geometry, coarse structure, uniform Roe algebra, uniformly discrete, locally finite
Received by editor(s): October 17, 2019
Received by editor(s) in revised form: April 15, 2020
Published electronically: November 3, 2020
Additional Notes: The first author was supported by York Science Research Fellowship.
Both authors were partially supported by IF’s NSERC grant.
Article copyright: © Copyright 2020 American Mathematical Society