The rigidity problem for uniform Roe algebras of uniformly locally finite coarse spaces
Baudier, Braga, Farah, Khukhro, Vignati and Willett proved that the answer is yes for uniformly locally finite metric spaces.
Problem. Let $(X, \mathcal E)$ and $(Y, \mathcal E')$ be uniformly locally finite coarse spaces whose uniform Roe algebras $C^*_u(X, \mathcal E)$ and $C^*_u(Y, \mathcal E')$ are isomorphic. Must $(X, \mathcal E)$ and $(Y, \mathcal E')$ be coarsely equivalent?
Source: B. M. Braga and I. Farah (2021) and Braga, Farah and A. Vignati (2022), Problem 1.1, after J. Špakula and R. Willett (2013) for metric spaces; listed as B.1 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).
Imported from math.uwaterloo.ca/~j37peter/problems.html.
Classification
- Primary
- OA — Operator Algebras
- Secondary
- MG
- Keywords
- coarse geometry, Peterson's operator algebra problems, rigidity, uniform Roe algebras
References
@article{braga2021,
author = {Braga, Bruno M. and Farah, Ilijas},
title = {On the rigidity of uniform {R}oe algebras over uniformly locally finite coarse spaces},
journal = {Trans. Amer. Math. Soc.},
volume = {374},
number = {2},
pages = {1007--1040},
year = {2021},
doi = {10.1090/tran/8180},
}
@article{braga2022,
author = {Braga, Bruno M. and Farah, Ilijas and Vignati, Alessandro},
title = {General uniform {R}oe algebra rigidity},
journal = {Ann. Inst. Fourier},
volume = {72},
number = {1},
pages = {301--337},
year = {2022},
doi = {10.5802/aif.3461},
}
@misc{peterson-b1,
author = {Peterson, Jesse},
title = {Open problems in operator algebras},
howpublished = {Web page},
year = {2026},
url = {https://www.math.uwaterloo.ca/~j37peter/problems.html},
note = {Problem B.1; version of 15 May 2026},
}
@article{spakula2013,
author = {{\v{S}}pakula, J{\'a}n and Willett, Rufus},
title = {On rigidity of {R}oe algebras},
journal = {Adv. Math.},
volume = {249},
pages = {289--310},
year = {2013},
doi = {10.1016/j.aim.2013.09.006},
}
@article{baudier2022,
author = {Baudier, Florent P. and Braga, Bruno M. and Farah, Ilijas and Khukhro, Ana and Vignati, Alessandro and Willett, Rufus},
title = {Uniform {R}oe algebras of uniformly locally finite metric spaces are rigid},
journal = {Invent. Math.},
volume = {230},
number = {3},
pages = {1071--1100},
year = {2022},
doi = {10.1007/s00222-022-01140-x},
}
@misc{vignati2026,
author = {Vignati, Alessandro},
title = {The rigidity problem for uniform {R}oe algebras},
year = {2026},
eprint = {2602.01998},
archivePrefix = {arXiv},
}
Resolution claims
Let $(X,\mathcal E)$ and $(Y,\mathcal F)$ be uniformly locally finite coarse spaces. We prove that every $C^*$-algebra isomorphism $C_u^*(X,\mathcal E)\cong C_u^*(Y,\mathcal F)$ forces $(X,\mathcal E)$ and $(Y,\mathcal F)$ to be bijectively coarsely equivalent. This completely resolves the isomorphism rigidity problem for uniform Roe algebras over arbitrary uniformly locally finite coarse spaces.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-29 | Needs verification → Solution claimed | — | Recorded here: arXiv:2607.15096v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01066, "The rigidity problem for uniform Roe algebras of uniformly locally finite coarse spaces", clopen.solutions.