OP-00997 Open

Peterson's conjecture on property (T) subfactors of ultrapowers of II₁ factors with the Haagerup property

Problem. Let $N$ be a II$_1$ factor with the Haagerup property, $\mathcal U$ a free ultrafilter on $\mathbb N$, and $M \subseteq N^{\mathcal U}$ a subfactor with property (T). Are there finite-dimensional von Neumann subalgebras $N_n \subseteq N$ with $M \subseteq \prod_{\mathcal U} N_n \subseteq N^{\mathcal U}$?

Source: J. Peterson, AimPL problem list Classification of group von Neumann algebras (2018), Problem 1.1, as an analogue of Shalom's theorem for groups; listed as U.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
Keywords: Haagerup property, Peterson's operator algebra problems, property (T), ultrapowers
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{aimpl2018-u5,
  title        = {{AimPL}: Classification of group von {N}eumann algebras},
  howpublished = {AIM problem list},
  year         = {2018},
  url          = {http://aimpl.org/groupvonneumann},
  note         = {Problem 1.1},
}
@misc{peterson-u5,
  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.5; version of 15 May 2026},
}
@misc{peterson2026-u5,
  author        = {Peterson, Jesse},
  title         = {On ultraproduct approximations and property {(T)} factors},
  year          = {2026},
  eprint        = {2605.16669},
  archivePrefix = {arXiv},
  note          = {v2, 1 Oct 2026; Theorem 1.3 proves the case N = L(F_n) when the relative commutant of M is a factor},
}
@misc{kunnawalkam2025-u5,
  author        = {Kunnawalkam Elayavalli, Srivatsav},
  title         = {50 open problems: ultraproduct {II}$_1$ factors},
  year          = {2025},
  eprint        = {2511.20377},
  archivePrefix = {arXiv},
  note          = {Problem 12; v3, 25 Aug 2026},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-00997, "Peterson's conjecture on property (T) subfactors of ultrapowers of II₁ factors with the Haagerup property", clopen.solutions.