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

HTML articles powered by AMS MathViewer

- 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.## References

- Goulnara Arzhantseva, Erik Guentner, and Ján Špakula,
*Coarse non-amenability and coarse embeddings*, Geom. Funct. Anal.**22**(2012), no. 1, 22–36. MR**2899681**, DOI 10.1007/s00039-012-0145-z - B. M. Braga, Y.-C. Chung, and K. Li,
*Coarse Baum-Connes conjecture and rigidity for Roe algebras*. J. Funct. Anal. 279 (2020), no. 9, 108728, 21 pp. DOI: 10.1016/j.jfa.2020.108728. - B. Braga and I. Farah,
*On the rigidity of uniform Roe algebras over uniformly locally finite coarse spaces*, ArXiv e-prints. - B. M. Braga, I. Farah, and A. Vignati,
*Uniform Roe coronas*, arXiv:1810.07789. - Bruno M. Braga, Ilijas Farah, and Alessandro Vignati,
*Embeddings of uniform Roe algebras*, Comm. Math. Phys.**377**(2020), no. 3, 1853–1882. MR**4121613**, DOI 10.1007/s00220-019-03539-9 - J. Brodzki, G. A. Niblo, and N. J. Wright,
*Property A, partial translation structures, and uniform embeddings in groups*, J. Lond. Math. Soc. (2)**76**(2007), no. 2, 479–497. MR**2363428**, DOI 10.1112/jlms/jdm066 - B. M. Braga, A. Vignati,
*On the uniform Roe algebra as a Banach algebra and embeddings of $\ell _p$ uniform Roe algebras*, Bulletin of the London Mathematical Society (2020) 1–18. DOI:10.1112/blms.12366. - Yeong Chyuan Chung and Kang Li,
*Rigidity of $\ell ^p$ Roe-type algebras*, Bull. Lond. Math. Soc.**50**(2018), no. 6, 1056–1070. MR**3891943**, DOI 10.1112/blms.12201 - Xiaoman Chen and Qin Wang,
*Ideal structure of uniform Roe algebras of coarse spaces*, J. Funct. Anal.**216**(2004), no. 1, 191–211. MR**2091361**, DOI 10.1016/j.jfa.2003.11.015 - Xiaoman Chen and Qin Wang,
*Ghost ideals in uniform Roe algebras of coarse spaces*, Arch. Math. (Basel)**84**(2005), no. 6, 519–526. MR**2148492**, DOI 10.1007/s00013-005-1189-1 - Eske Ellen Ewert and Ralf Meyer,
*Coarse geometry and topological phases*, Comm. Math. Phys.**366**(2019), no. 3, 1069–1098. MR**3927086**, DOI 10.1007/s00220-019-03303-z - Martin Finn-Sell,
*Fibred coarse embeddings, a-T-menability and the coarse analogue of the Novikov conjecture*, J. Funct. Anal.**267**(2014), no. 10, 3758–3782. MR**3266245**, DOI 10.1016/j.jfa.2014.09.012 - Marshall Hall Jr.,
*Combinatorial theory*, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1967. MR**0224481** - N. Higson, V. Lafforgue, and G. Skandalis,
*Counterexamples to the Baum-Connes conjecture*, Geom. Funct. Anal.**12**(2002), no. 2, 330–354. MR**1911663**, DOI 10.1007/s00039-002-8249-5 - Yosuke Kubota,
*Controlled topological phases and bulk-edge correspondence*, Comm. Math. Phys.**349**(2017), no. 2, 493–525. MR**3594362**, DOI 10.1007/s00220-016-2699-3 - Kenneth Kunen,
*Set theory*, Studies in Logic (London), vol. 34, College Publications, London, 2011. MR**2905394** - Kang Li and Hung-Chang Liao,
*Classification of uniform Roe algebras of locally finite groups*, J. Operator Theory**80**(2018), no. 1, 25–46. MR**3835447**, DOI 10.7900/jot.2017may23.2163 - Kang Li and Rufus Willett,
*Low-dimensional properties of uniform Roe algebras*, J. Lond. Math. Soc. (2)**97**(2018), no. 1, 98–124. MR**3764069**, DOI 10.1112/jlms.12100 - Hans Jürgen Prömel and Bernd Voigt,
*Canonical forms of Borel-measurable mappings $\Delta \colon \ [\omega ]^\omega \to \textbf {R}$*, J. Combin. Theory Ser. A**40**(1985), no. 2, 409–417. MR**814423**, DOI 10.1016/0097-3165(85)90099-8 - John Roe,
*An index theorem on open manifolds. I, II*, J. Differential Geom.**27**(1988), no. 1, 87–113, 115–136. MR**918459** - John Roe,
*Coarse cohomology and index theory on complete Riemannian manifolds*, Mem. Amer. Math. Soc.**104**(1993), no. 497, x+90. MR**1147350**, DOI 10.1090/memo/0497 - John Roe,
*Lectures on coarse geometry*, University Lecture Series, vol. 31, American Mathematical Society, Providence, RI, 2003. MR**2007488**, DOI 10.1090/ulect/031 - C. Rosendal,
*Coarse Geometry of Topological Groups*, Book manuscript, version of 2018, http://homepages.math.uic.edu/~rosendal/PapersWebsite/Coarse-Geometry-Book23.pdf. - John Roe and Rufus Willett,
*Ghostbusting and property A*, J. Funct. Anal.**266**(2014), no. 3, 1674–1684. MR**3146831**, DOI 10.1016/j.jfa.2013.07.004 - H. Sako,
*Finite-dimensional approximation properties for uniform Roe algebras*, arXiv e-prints, page arXiv:1212.5900, December 2012. - H. Sako,
*Property A for coarse spaces*, ArXiv e-prints, March 2013. - Saharon Shelah,
*Cardinal arithmetic for skeptics*, Bull. Amer. Math. Soc. (N.S.)**26**(1992), no. 2, 197–210. MR**1112424**, DOI 10.1090/S0273-0979-1992-00261-6 - G. Skandalis, J. L. Tu, and G. Yu,
*The coarse Baum-Connes conjecture and groupoids*, Topology**41**(2002), no. 4, 807–834. MR**1905840**, DOI 10.1016/S0040-9383(01)00004-0 - Ján Špakula and Rufus Willett,
*On rigidity of Roe algebras*, Adv. Math.**249**(2013), 289–310. MR**3116573**, DOI 10.1016/j.aim.2013.09.006 - S. White and R. Willett,
*Cartan subalgebras of uniform Roe algebras*, Groups, Geometry, and Dynamics, to appear 2019. - Wilhelm Winter and Joachim Zacharias,
*The nuclear dimension of $C^\ast$-algebras*, Adv. Math.**224**(2010), no. 2, 461–498. MR**2609012**, DOI 10.1016/j.aim.2009.12.005 - Guoliang Yu,
*The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space*, Invent. Math.**139**(2000), no. 1, 201–240. MR**1728880**, DOI 10.1007/s002229900032

## Additional Information

**B. M. Braga**- Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, M3J IP3, Canada
- MR Author ID: 1094570
- Email: demendoncabraga@gmail.com
**I. Farah**- Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, M3J IP3, Canada
- MR Author ID: 350129
- ORCID: 0000-0001-7703-6931
- Email: ifarah@mathstat.yorku.ca
- 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. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 1007-1040 - MSC (2010): Primary 46L80, 46L85, 51K05
- DOI: https://doi.org/10.1090/tran/8180
- MathSciNet review: 4196385