Authors: Baudier, Florent P.
Braga, Bruno M.
Farah, Ilijas 
Khukhro, Ana
Vignati, Alessandro
Willett, Rufus
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Uniform Roe algebras of uniformly locally finite metric spaces are rigid
Journal: Inventiones Mathematicae
Volume: 230
First page: 1071
Last page: 1100
Issue Date: 2022
Rank: ~M21a
ISSN: 0020-9910
DOI: 10.1007/s00222-022-01140-x
We show that if X and Y are uniformly locally finite metric spaces whose uniform Roe algebras, Cu∗(X) and Cu∗(Y), are isomorphic as C ∗-algebras, then X and Y are coarsely equivalent metric spaces. Moreover, we show that coarse equivalence between X and Y is equivalent to Morita equivalence between Cu∗(X) and Cu∗(Y). As an application, we obtain that if Γ and Λ are finitely generated groups, then the crossed products ℓ∞(Γ) ⋊ rΓ and ℓ∞(Λ) ⋊ rΛ are isomorphic if and only if Γ and Λ are bi-Lipschitz equivalent.
Publisher: Springer Link

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