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

Braga, B.; Farah, I.

doi:10.1090/tran/8180

On the rigidity of uniform Roe algebras over uniformly locally finite coarse spaces
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by B. M. Braga and I. Farah PDF

Trans. Amer. Math. Soc. 374 (2021), 1007-1040 Request permission

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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