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Do reduced group C*-algebras remember torsion-free groups?

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Problem. Let $\Gamma$ and $\Lambda$ be torsion-free groups with $C^*_r(\Gamma) \cong C^*_r(\Lambda)$. Must $\Gamma$ and $\Lambda$ be isomorphic?

Source: Ioana, Popa and Vaes (2013), who note it seems unknown; credited to Popa and Vaes and listed as S.5 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
GR
Keywords
C*-superrigidity, Peterson's operator algebra problems, reduced group C*-algebras, torsion-free groups
Literature checked
Oct. 26, 2026 by clemens-admin

References

@article{ioana2013,
  author  = {Ioana, Adrian and Popa, Sorin and Vaes, Stefaan},
  title   = {A class of superrigid group von {N}eumann algebras},
  journal = {Ann. of Math.},
  volume  = {178},
  number  = {1},
  pages   = {231--286},
  year    = {2013},
  doi     = {10.4007/annals.2013.178.1.4},
}
@misc{peterson-s5,
  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.5; version of 15 May 2026},
}
@misc{ariza2026,
  author        = {Ariza Mej{\'i}a, Juan Felipe and Chifan, Ionu{\c t} and Fern{\'a}ndez Quero, Adriana},
  title         = {Non-amenable {$C^*$}-superrigid groups that are not {$W^*$}-superrigid},
  year          = {2026},
  eprint        = {2602.05290},
  archivePrefix = {arXiv},
}
@article{chifan2023,
  author  = {Chifan, Ionu{\c t} and Diaz-Arias, Alec and Drimbe, Daniel},
  title   = {New examples of {$W^*$} and {$C^*$}-superrigid groups},
  journal = {Adv. Math.},
  volume  = {412},
  pages   = {108797},
  year    = {2023},
  doi     = {10.1016/j.aim.2022.108797},
}
@article{omland2020,
  author  = {Omland, Tron},
  title   = {Free nilpotent groups are {$C^*$}-superrigid},
  journal = {Proc. Amer. Math. Soc.},
  volume  = {148},
  number  = {1},
  pages   = {283--287},
  year    = {2020},
  doi     = {10.1090/proc/14678},
}
@article{eckhardt2018,
  author  = {Eckhardt, Caleb and Raum, Sven},
  title   = {{$C^*$}-superrigidity of 2-step nilpotent groups},
  journal = {Adv. Math.},
  volume  = {338},
  pages   = {175--195},
  year    = {2018},
  doi     = {10.1016/j.aim.2018.09.008},
}

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OP-00964, "Do reduced group C*-algebras remember torsion-free groups?", clopen.solutions.