Unique Cartan subalgebras for groups with nonzero first ℓ²-Betti number
Popa and Vaes proved that for every weakly amenable group $\Gamma$ with $\beta_1^{(2)}(\Gamma) > 0$, every free ergodic probability measure preserving action gives a II$_1$ factor with a unique Cartan subalgebra up to unitary conjugacy.
Problem. Let $\Gamma$ be a group with $\beta_1^{(2)}(\Gamma) \ne 0$. Does every free ergodic probability measure preserving action $\Gamma \curvearrowright (X, \mu)$ give a II$_1$ factor $L^\infty(X) \rtimes \Gamma$ with a unique Cartan subalgebra up to unitary conjugacy?
Source: Conjecture of S. Popa and S. Vaes for the first ℓ²-Betti number: S. Popa, Some open problems in W*-rigidity (2013); AimPL problem list Classification of group von Neumann algebras (2018), Problem 3.2; Popa–Vaes, Acta Math. 212 (2014). Problem I in A. Ioana, ICM 2018; listed without attribution as C.4 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
- DS, GR
- Keywords
- Cartan subalgebras, C-rigidity, ℓ²-Betti numbers, Peterson's operator algebra problems
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@misc{popa2013problems-c4,
author = {Popa, Sorin},
title = {Some open problems in {W}$^*$-rigidity},
howpublished = {Slides, Paris, June 2013},
year = {2013},
url = {https://www.math.ucla.edu/~popa/ProblemsJune2013.pdf},
}
@misc{aimpl2018-c4,
title = {{AimPL}: Classification of group von {N}eumann algebras},
howpublished = {AIM problem list},
year = {2018},
url = {http://aimpl.org/groupvonneumann},
note = {Problem 3.2},
}
@article{popavaes2014acta-c4,
author = {Popa, Sorin and Vaes, Stefaan},
title = {Unique {C}artan decomposition for {II}$_1$ factors arising from arbitrary actions of free groups},
journal = {Acta Math.},
volume = {212},
number = {1},
pages = {141--198},
year = {2014},
doi = {10.1007/s11511-014-0110-9},
}
@misc{peterson-c4,
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 C.4; version of 15 May 2026},
}
@inproceedings{ioana2018-c4,
author = {Ioana, Adrian},
title = {Rigidity for von {N}eumann algebras},
booktitle = {Proceedings of the International Congress of Mathematicians (Rio de Janeiro 2018), Vol. III},
publisher = {World Sci. Publ.},
pages = {1639--1672},
year = {2018},
doi = {10.1142/9789813272880_0111},
}
@article{chifan2013-c4,
author = {Chifan, Ionut and Peterson, Jesse},
title = {Some unique group-measure space decomposition results},
journal = {Duke Math. J.},
volume = {162},
number = {11},
pages = {1923--1966},
year = {2013},
doi = {10.1215/00127094-2331230},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01006, "Unique Cartan subalgebras for groups with nonzero first ℓ²-Betti number", clopen.solutions.