OP-00999 Open

Existential embedding of L(F₂) into a matrix ultraproduct

Problem. Is there an existential embedding of $L(\mathbb F_2)$ into $\prod_{n \to \mathcal U} \mathbb M_n(\mathbb C)$, for a free ultrafilter $\mathcal U$ on $\mathbb N$?

Source: S. Kunnawalkam Elayavalli, 50 open problems: ultraproduct II₁ factors (2025), Problem 13; listed as U.10 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, existential embeddings, free group factors, Peterson's operator algebra problems, ultraproducts
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{kunnawalkam2025-u10,
  author        = {Kunnawalkam Elayavalli, Srivatsav},
  title         = {50 open problems: ultraproduct {II}$_1$ factors},
  year          = {2025},
  eprint        = {2511.20377},
  archivePrefix = {arXiv},
  note          = {Problem 13; v3, 25 Aug 2026},
}
@misc{peterson-u10,
  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.10; version of 15 May 2026},
}
@misc{peterson2026-u10,
  author        = {Peterson, Jesse},
  title         = {On ultraproduct approximations and property {(T)} factors},
  year          = {2026},
  eprint        = {2605.16669},
  archivePrefix = {arXiv},
  note          = {v2, 1 Oct 2026},
}
@misc{hayes2026random,
  author        = {Hayes, Ben and Jekel, David and Kunnawalkam Elayavalli, Srivatsav},
  title         = {Questions on the structure of random embeddings of {$L(\mathbb F_2)$}},
  year          = {2026},
  eprint        = {2606.02985},
  archivePrefix = {arXiv},
  note          = {v1, 2 Jun 2026},
}
@article{jekel2026,
  author  = {Jekel, David and Kunnawalkam Elayavalli, Srivatsav},
  title   = {Upgraded free independence phenomena for random unitaries},
  journal = {Trans. Amer. Math. Soc. Ser. B},
  volume  = {13},
  pages   = {1--29},
  year    = {2026},
  doi     = {10.1090/btran/244},
}
@article{kunnawalkam2023,
  author  = {Kunnawalkam Elayavalli, Srivatsav},
  title   = {Remarks on the diagonal embedding and strong 1-boundedness},
  journal = {Doc. Math.},
  volume  = {28},
  number  = {3},
  pages   = {671--681},
  year    = {2023},
  doi     = {10.4171/DM/918},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-00999, "Existential embedding of L(F₂) into a matrix ultraproduct", clopen.solutions.