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Uniform Roe algebras of uniformly locally finite metric spaces are rigid

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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.

References

  1. Albiac, F., Kalton, N.J.: Topics in Banach Space Theory, vol. 233 of Graduate Texts in Mathematics. Springer, [Cham], second edition, (2016). With a foreword by Gilles Godefory

  2. Block, J., Weinberger, S.: Aperiodic tilings, positive scalar curvature and amenability of spaces. J. Am. Math. Soc. 5(4), 907–918 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Braga, B.M., Farah, I.: On the rigidity of uniform Roe algebras over uniformly locally finite coarse spaces. Trans. Am. Math. Soc. 374(2), 1007–1040 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  4. Braga, B.M., Farah, I., Vignati, A.: Uniform Roe coronas. Adv. Math. 389(107886), 35 (2021)

    MathSciNet  MATH  Google Scholar 

  5. Braga, B.M., Farah, I., Vignati, A.: General uniform Roe algebra rigidity. Ann. Inst. Fourier (Grenoble) 72(1), 301–337 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  6. Braga, B.M.: On Banach algebras of band-dominated operators and their order structure. J. Funct. Anal., 280(9):Paper No. 108958, 40 (2021)

  7. Braga, Bruno M., Vignati, Alessandro: On the uniform Roe algebra as a Banach algebra and embeddings of \(\ell _p\) uniform Roe algebras. Bull. Lond. Math. Soc. 52(5), 853–870 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brodzki, J., Cave, C., Li, K.: Exactness of locally compact second countable groups. Adv. Math. 312, 209–233 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brodzki, J., Niblo, G.A., Wright, N.J.: Property A, partial translation structures, and uniform embeddings in groups. J. Lond. Math. Soc. (2), 76(2), 479–497 (2007)

  10. Brown, L., Green, P., Rieffel, M.: Stable isomorphism and strong Morita equivalence of \(C^*\)-algebras. Pacific J. Math. 71(2), 349–363 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brown, N., Ozawa, N.: \(C^*\)-algebras and finite-dimensional approximations. Graduate Studies in Mathematics, vol. 88. American Mathematical Society, Providence, RI (2008)

  12. Cedzich, C., Gelb, T., Stahl, C., Velázquez, L., Werner, A.H., Werner, R.F.: Complete homotopy invariants for translation invariant symmetric quantum walks on a chain. Quantum 2, 95

  13. Chen, X., Tessera, R., Wang, X., Yu, G.: Metric sparsification and operator norm localization. Adv. Math. 218(5), 1496–1511 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, X., Wang, Q.: Ideal structure of uniform Roe algebras of coarse spaces. J. Funct. Anal. 216(1), 191–211 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chung, Y., Li, K.: Rigidity of \(\ell ^p\) Roe-type algebras. Bull. Lond. Math. Soc. 50(6), 1056–1070 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dixmier, J.: \({C^*}\)-Algebras. North Holland Publishing Company (1977)

  17. Elton, J., Hill, T.: A generalization of Lyapounov’s convexity theorem to measures with atoms. Proc. Am. Math. Soc. 99(2), 297–304 (1987)

    MathSciNet  MATH  Google Scholar 

  18. Engel, A.: Index theorems for uniformly elliptic operators. New York J. Math. 24, 543–587 (2018)

    MathSciNet  MATH  Google Scholar 

  19. Farah, I.: Combinatorial Set Theory and \({{\rm C}}^{\ast }\)-algebras. Springer Monographs in Mathematics. Springer (2019)

  20. Gromov, M.: Asymptotic invariants of infinite groups. In: Geometric Group Theory, Vol. 2 (Sussex, 1991), vol. 182 of London Mathematical Society Lecture Note Series, pp. 1–295. Cambridge Univ. Press, Cambridge (1993)

  21. Guentner, E., Kaminker, J.: Exactness and uniform embeddability of discrete groups. J. Lond. Math. Soc. (2), 70(3), 703–718 (2004)

  22. Kellerhals, J., Monod, N., Rørdam, M.: Non-supramenable groups acting on locally compact spaces. Doc. Math. 18, 1597–1626 (2013)

    MathSciNet  MATH  Google Scholar 

  23. Kubota, Y.: Controlled topological phases and bulk-edge correspondence. Commun. Math. Phys. 349(2), 493–525 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lance, E.C.: Hilbert \(C^*\)-modules, volume 210 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1995). A toolkit for operator algebraists

  25. Li, K., Špakula, J., Zhang, J.: Measured asymptotic expanders and rigidity for roe algebras (2021)

  26. Li, K., Willett, R.: Low-dimensional properties of uniform Roe algebras. J. Lond. Math. Soc. 97, 98–124 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liapounoff. A.: Sur les fonctions-vecteurs complètement additives. Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR], 4:465–478 (1940)

  28. Lindenstrauss, J.: A short proof of Liapounoff’s convexity theorem. J. Math. Mech. 15, 971–972 (1966)

    MathSciNet  MATH  Google Scholar 

  29. Lindenstrauss, J.: On James’s paper “Separable conjugate spaces.” Israel J. Math. 9, 279–284 (1971)

  30. Nowak, P., Yu, G.: Large Scale Geometry. EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich (2012)

  31. Rabinovich, V.S., Roch, S., Roe, J.: Fredholm indices of band-dominated operators on discrete groups. Integral Equ. Oper. Theory 49, 221–238 (2004)

    Article  MATH  Google Scholar 

  32. Roe, J.: An index theorem on open manifolds. I. J. Differ. Geom. 27, 87–113 (1988)

    MathSciNet  MATH  Google Scholar 

  33. Roe, J.: Lectures on Coarse Geometry. University Lecture Series, vol. 31. American Mathematical Society, Providence, RI (2003)

  34. Roe, J., Willett, R.: Ghostbusting and property A. J. Funct. Anal. 266(3), 1674–1684 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rørdam, M., Sierakowski, A.: Purely infinite \({C}^*\)-algebras arising from crossed products. Ergodic Theory Dyn. Syst. 32, 273–293 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sako, H.: Property A and the operator norm localization property for discrete metric spaces. J. Reine Angew. Math. 690, 207–216 (2014)

    MathSciNet  MATH  Google Scholar 

  37. Špakula, J.: Uniform \({K}\)-homology theory. J. Funct. Anal. 257(1), 88–121 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Špakula, J., Willett, R.: On rigidity of Roe algebras. Adv. Math. 249, 289–310 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. Špakula, J., Willett, R.: A metric approach to limit operators. Trans. Am. Math. Soc. 369, 263–308 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  40. Starr, R.: Quasi-equilibria in markets with non-convex preferences. Econometrica 37(1), 25–38 (1969)

    Article  MATH  Google Scholar 

  41. Špakula, J., Tikuisis, A.: Relative commutant picture of Roe algebras. arXiv:1707.04552 (2017)

  42. Špakula, J., Zhang, J.: Quasi-locality and property A. arXiv:1809.00532 (2018)

  43. White, S., Willett, R.: Cartan subalgebras in uniform Roe algebras. Groups Geom. Dyn. 14(3), 949–989 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  44. Whyte, K.: Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture. Duke Math. J. 99(1), 93–112 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  45. Willett, R.: Some notes on property A. In: Limits of Graphs in Group Theory and Computer Science, pp. 191–281. EPFL Press, Lausanne (2009)

  46. Yu, G.: The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math. 139(1), 201–240 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zhou, L.: A simple proof of the Shapley-Folkman theorem. Econom. Theory 3(2), 371–372 (1993)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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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Authors and Affiliations

  1. Department of Mathematics, Texas A &M University, College Station, TX, 77843-3368, USA

    Florent P. Baudier

  2. PUC-Rio, Rua Marquês de São Vicente 225, Rio de Janeiro, RJ, Brazil

    Bruno M. Braga

  3. Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, ON, M3J 1P3, Canada

    Ilijas Farah

  4. Matematički Institut SANU, Kneza Mihaila 36, p.p. 367, 11 000, Beograd, Serbia

    Ilijas Farah

  5. Murray Edwards College, University of Cambridge, Huntingdon Road, Cambridge, CB3 0DF, UK

    Ana Khukhro

  6. Institut de Mathématiques de Jussieu (IMJ-PRG), Université Paris Cité, Bâtiment Sophie Germain, 8 Place Aurélie Nemours, 75013, Paris, France

    Alessandro Vignati

  7. University of Hawai’i at Mānoa, 2565 McCarthy Mall, Keller 401A, Honolulu, HI, 96822, USA

    Rufus Willett

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  1. Florent P. Baudier
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  2. Bruno M. Braga
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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. 230, 1071–1100 (2022). https://doi.org/10.1007/s00222-022-01140-x

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