OP-00995 Open

Fundamental groups of ultrapowers of II₁ factors

Problem. Is there a II$_1$ factor $M$ and a free ultrafilter $\mathcal U$ on $\mathbb N$ such that the fundamental group $\mathcal F(M^{\mathcal U})$ is not $\mathbb R_+^*$?

Source: AimPL problem list Classification of group von Neumann algebras (2018), Problem 1.2; listed as U.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: LO
Keywords: continuous model theory, fundamental group of II₁ factors, Peterson's operator algebra problems, ultrapowers
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{aimpl2018-u1,
  title        = {{AimPL}: Classification of group von {N}eumann algebras},
  howpublished = {AIM problem list},
  year         = {2018},
  url          = {http://aimpl.org/groupvonneumann},
  note         = {Problem 1.2},
}
@misc{peterson-u1,
  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 U.1; version of 15 May 2026},
}
@misc{kunnawalkam2025-u1,
  author        = {Kunnawalkam Elayavalli, Srivatsav},
  title         = {50 open problems: ultraproduct {II}$_1$ factors},
  year          = {2025},
  eprint        = {2511.20377},
  archivePrefix = {arXiv},
  note          = {Problem 48; v3, 25 Aug 2026},
}
@incollection{goldbring2023survey,
  author        = {Goldbring, Isaac and Hart, Bradd},
  title         = {A survey on the model theory of tracial von {N}eumann algebras},
  booktitle     = {Model theory of operator algebras},
  series        = {De Gruyter Ser. Log. Appl.},
  volume        = {11},
  publisher     = {De Gruyter},
  pages         = {133--158},
  year          = {2023},
  doi           = {10.1515/9783110768282-004},
  eprint        = {2210.15508},
  archivePrefix = {arXiv},
}
@misc{peterson2026-u1,
  author        = {Peterson, Jesse},
  title         = {On ultraproduct approximations and property {(T)} factors},
  year          = {2026},
  eprint        = {2605.16669},
  archivePrefix = {arXiv},
  note          = {v2, 1 Oct 2026},
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00995, "Fundamental groups of ultrapowers of II₁ factors", clopen.solutions.