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Uniform Roe algebras of uniformly locally finite metric spaces are rigid

Abstract

We show that if X and Y are uniformly locally finite metric spaces whose uniform Roe algebras, \(\mathrm {C}^*_u(X)\) and \(\mathrm {C}^*_u(Y)\), are isomorphic as \(\mathrm {C}^*\)-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 \(\mathrm {C}^*_u(X)\) and \(\mathrm {C}^*_u(Y)\). As an application, we obtain that if \(\Gamma \) and \(\Lambda \) are finitely generated groups, then the crossed products \(\ell _\infty (\Gamma )\rtimes _r\Gamma \) and \( \ell _\infty (\Lambda )\rtimes _r\Lambda \) are isomorphic if and only if \(\Gamma \) and \(\Lambda \) are bi-Lipschitz equivalent.

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Notes

  1. Also called bounded geometry metric spaces in the literature.

  2. Metric spaces \((X,d_X)\) and \((Y,d_Y)\) are bi-Lipschitz equivalent if there is a bijection \(f:X \rightarrow Y\) such that f and \(f^{-1}\) are Lipschitz.

  3. For the set theorist reader, this result is a theorem in \(\mathrm {ZFC}+\mathrm {OCA}_{\mathrm {T}}+\mathrm {MA}_{\aleph _1}\).

  4. This was formalized as rigidity of a \(*\)-isomorphism in [3, p. 1008].

  5. Here we use the following standard notation: for \(S\subseteq X\), we let \(\chi _{S} :=\mathrm {SOT}\text {-}\sum _{x\in S}e_{xx}\), i.e., \(\chi _S\) is the operator on \(\ell _2(X)\) that projects onto the coordinates indexed by S.

  6. For a proof that \(e_n^*\) is extends to a well-defined bounded linear functional, see for example [1, Theorem 1.1.3].

  7. Any countable group admits such a metric, which is moreover unique up to bijective coarse equivalence (e.g., [45, Proposition 2.3.3]).

  8. We will actually only need that \(\Vert (\chi _{C}\otimes 1_H)p_B(\chi _{D}\otimes 1_H)\Vert \leqslant \varepsilon \) for all \(C,D\subseteq X\) with \(d(C,D)> r\) and all \(A\subseteq {\mathbb {N}}\).

  9. See [24, Chapter 7] for a shorter proof.

  10. In this context, countably generated means that there is a countable collection S of subsets of \(X \times X\) such that \({\mathcal {E}}\) is the intersection of all coarse structures containing S, and connected means that \(\{(x,y)\}\in {\mathcal {E}}\) for all \(x,y\in X\). The connectedness condition in the metric setting means that metrics are not allowed to take infinite values.

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Acknowledgements

This paper was written under the auspices of the American Institute of Mathematics (AIM) SQuaREs program and as part of the ‘Expanders, ghosts, and Roe algebras’ SQuaRE project. F. B., B. M. B. and R. W. were partially supported by the US National Science Foundation under the grants DMS-1800322 and DMS-2055604, DMS-2054860, and DMS-1901522, respectively. I. F. was partially supported by NSERC. A. V. is supported by an ‘Emergence en Recherche’ IdeX grant from the Université de Paris and an ANR grant (ANR-17-CE40-0026). Last, but not least, we are indebted to the anonymous referee for a thoughtful report: in particular, the referee suggested that Lemma 2.1 can be proved using [29], and provided a better insight into the proof of our Lemma 3.2 that resulted in substantial simplification in this, and also of Lemma 4.3.

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Baudier, F.P., Braga, B.M., Farah, I. et al. Uniform Roe algebras of uniformly locally finite metric spaces are rigid. Invent. math. (2022). https://doi.org/10.1007/s00222-022-01140-x

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