OP-00998 Open

Embeddings into matrix ultraproducts with trivial relative commutant

Problem. Let $M$ be a separable full II$_1$ factor that embeds into $\prod_{n \to \mathcal U} \mathbb M_n(\mathbb C)$ for a free ultrafilter $\mathcal U$ on $\mathbb N$. Is there such an embedding whose relative commutant is trivial? Is there one when $M$ has property (T)?

Source: Credited to A. Ioana and listed as U.9 in J. Peterson, Open problems in operator algebras (version of 15 May 2026); credited to Ioana and Popa as Problem 2 of S. Kunnawalkam Elayavalli, 50 open problems: ultraproduct II₁ factors (2025).

Imported from math.uwaterloo.ca/~j37peter/problems.html.

Classification

Primary: OA — Operator Algebras
Secondary: LO
Keywords: full factors, matrix algebras, Peterson's operator algebra problems, relative commutants, ultraproducts
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{peterson-u9,
  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.9; version of 15 May 2026},
}
@misc{kunnawalkam2025-u9,
  author        = {Kunnawalkam Elayavalli, Srivatsav},
  title         = {50 open problems: ultraproduct {II}$_1$ factors},
  year          = {2025},
  eprint        = {2511.20377},
  archivePrefix = {arXiv},
  note          = {Problem 2; v3, 25 Aug 2026},
}
@misc{peterson2026-u9,
  author        = {Peterson, Jesse},
  title         = {On ultraproduct approximations and property {(T)} factors},
  year          = {2026},
  eprint        = {2605.16669},
  archivePrefix = {arXiv},
  note          = {v2, 1 Oct 2026; Proposition 7.8 treats L(SL_3(Z))},
}
@article{brown2011,
  author  = {Brown, Nathanial P.},
  title   = {Topological dynamical systems associated to {II}$_1$-factors},
  journal = {Adv. Math.},
  volume  = {227},
  number  = {4},
  pages   = {1665--1699},
  year    = {2011},
  doi     = {10.1016/j.aim.2011.04.003},
}
@article{hayes2018,
  author  = {Hayes, Ben},
  title   = {1-bounded entropy and regularity problems in von {N}eumann algebras},
  journal = {Int. Math. Res. Not. IMRN},
  year    = {2018},
  number  = {1},
  pages   = {57--137},
  doi     = {10.1093/imrn/rnw237},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-00998, "Embeddings into matrix ultraproducts with trivial relative commutant", clopen.solutions.