Is the reduced C*-algebra of the free group on countably infinitely many generators finitely generated?
Let $\mathbb F_\infty$ be the free group on countably infinitely many generators.
Problem. Is $C^*_r(\mathbb F_\infty)$ finitely generated as a C$^*$-algebra?
Source: Listed without attribution as S.6 in J. Peterson, Open problems in operator algebras (version of 15 May 2026); discussed by C. Schafhauser, A. Tikuisis and S. White, Nuclear C*-algebras: 99 problems (2025).
Imported from math.uwaterloo.ca/~j37peter/problems.html.
Classification
- Primary
- OA — Operator Algebras
- Keywords
- free groups, generator problem, Peterson's operator algebra problems, reduced group C*-algebras
References
@misc{peterson-s6,
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 S.6; version of 15 May 2026},
}
@misc{schafhauser2025,
author = {Schafhauser, Christopher and Tikuisis, Aaron and White, Stuart},
title = {Nuclear {C}$^*$-algebras: 99 problems},
year = {2025},
eprint = {2506.10902},
archivePrefix = {arXiv},
note = {v2, 8 May 2026},
}
@article{thiel2014,
author = {Thiel, Hannes and Winter, Wilhelm},
title = {The generator problem for {$\mathcal Z$}-stable {$C^*$}-algebras},
journal = {Trans. Amer. Math. Soc.},
volume = {366},
number = {5},
pages = {2327--2343},
year = {2014},
doi = {10.1090/S0002-9947-2014-06013-3},
}
Resolution claims
Let $A$ be a separable unital $\mathrm C^*$-algebra with nonempty tracial state space and strict comparison by ordinary tracial states. If $\mathcal Z$ embeds unitally into $A$, then $A$ is singly generated. In particular, every simple separable unital $\mathrm C^*$-algebra $A$ with $\mathrm T(A)\ne\varnothing$ and strict comparison is singly generated.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-29 | Needs verification → Solution claimed | — | Recorded here: arXiv:2610.17411v2. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01063, "Is the reduced C*-algebra of the free group on countably infinitely many generators finitely generated?", clopen.solutions.