OP-01051 Solution claimed

Do L(F₂) and L(F₃) have isomorphic ultrapowers?

Problem. Is there a free ultrafilter $\mathcal U$ on $\mathbb N$ such that $L(\mathbb F_2)^{\mathcal U} \cong L(\mathbb F_3)^{\mathcal U}$?

Source: The model-theoretic form of the free group factor problem: I. Goldbring and B. Hart's survey (2023), and S. Kunnawalkam Elayavalli, 50 open problems: ultraproduct II₁ factors (2025), Problem 36; listed as U.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: LO
Keywords: elementary equivalence, free group factors, Peterson's operator algebra problems, ultrapowers

References

@misc{peterson-u4,
  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.4; version of 15 May 2026},
}
@incollection{goldbring2023-u4,
  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{kunnawalkam2025-u4,
  author        = {Kunnawalkam Elayavalli, Srivatsav},
  title         = {50 open problems: ultraproduct {II}$_1$ factors},
  year          = {2025},
  eprint        = {2511.20377},
  archivePrefix = {arXiv},
  note          = {Problem 36; v3, 25 Aug 2026},
}
@article{goldbringpi2025,
  author  = {Goldbring, Isaac and Pi, Jennifer},
  title   = {On the first-order free group factor elementary equivalence},
  journal = {J. Operator Theory},
  volume  = {94},
  number  = {1},
  pages   = {129--150},
  year    = {2025},
  doi     = {10.7900/jot.2023sep11.246},
}

Resolution claims

An isomorphism of the free group factors v1 GitHub ↗

OpenAI

We solve the free group factor isomorphism problem affirmatively by proving that $L(\F_2)$ and $L(\F_3)$ are isomorphic as tracial von Neumann algebras. The classical free group factor alternative then implies that all interpolated free group factors, including $L(\F_\infty)$, are isomorphic and have fundamental group $\R_{>0}$.

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-29	Needs verification → Solution claimed	—	Recorded here: math @ 7ef5c0c, build/main.tex.

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-01051, "Do L(F₂) and L(F₃) have isomorphic ultrapowers?", clopen.solutions.