OP-00971 Open

Popa's factorial commutant embedding problem

Let $R$ be the hyperfinite II$_1$ factor and $\mathcal U$ a free ultrafilter on $\mathbb N$.

Problem. Let $M$ be a separable II$_1$ factor that embeds into $R^{\mathcal U}$. Is there an embedding $M \hookrightarrow R^{\mathcal U}$ whose relative commutant $M' \cap R^{\mathcal U}$ is a factor?

Source: S. Popa, Independence properties in subalgebras of ultraproduct II₁ factors (2014); listed as U.6 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: hyperfinite II₁ factor, Peterson's operator algebra problems, relative commutants, ultrapowers
Literature checked: Oct. 26, 2026 by clemens-admin

References

@article{popa2014,
  author  = {Popa, Sorin},
  title   = {Independence properties in subalgebras of ultraproduct {II}$_1$ factors},
  journal = {J. Funct. Anal.},
  volume  = {266},
  number  = {9},
  pages   = {5818--5846},
  year    = {2014},
  doi     = {10.1016/j.jfa.2014.02.004},
}
@misc{peterson-u6,
  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.6; version of 15 May 2026},
}
@article{goldbring2020,
  author  = {Goldbring, Isaac},
  title   = {On {P}opa's factorial commutant embedding problem},
  journal = {Proc. Amer. Math. Soc.},
  volume  = {148},
  number  = {11},
  pages   = {5007--5012},
  year    = {2020},
  doi     = {10.1090/proc/15141},
}
@article{atkinson2022,
  author  = {Atkinson, Scott and Goldbring, Isaac and Kunnawalkam Elayavalli, Srivatsav},
  title   = {Factorial relative commutants and the generalized {J}ung property for {II}$_1$ factors},
  journal = {Adv. Math.},
  volume  = {396},
  pages   = {108107},
  year    = {2022},
  doi     = {10.1016/j.aim.2021.108107},
}
@article{goldbringhart2020,
  author  = {Goldbring, Isaac and Hart, Bradd},
  title   = {Properties expressible in small fragments of the theory of the hyperfinite {II}$_1$ factor},
  journal = {Confluentes Math.},
  volume  = {12},
  number  = {2},
  pages   = {37--47},
  year    = {2020},
  doi     = {10.5802/cml.67},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-00971, "Popa's factorial commutant embedding problem", clopen.solutions.