OP-00996 Open

A II₁ factor whose ultrapowers differ from those of its opposite

Problem. Is there a II$_1$ factor $M$ such that $M^{\mathcal U} \not\cong (M^{\mathrm{op}})^{\mathcal U}$ for every free ultrafilter $\mathcal U$ on $\mathbb N$?

Source: Raised by I. Goldbring and T. Sinclair (2015) and in a remark to Problem 1.5 of the AimPL problem list Classification of group von Neumann algebras (2018); listed as U.3 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, opposite algebras, Peterson's operator algebra problems, ultrapowers
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{aimpl2018-u3,
  title        = {{AimPL}: Classification of group von {N}eumann algebras},
  howpublished = {AIM problem list},
  year         = {2018},
  url          = {http://aimpl.org/groupvonneumann},
  note         = {Remark to Problem 1.5},
}
@article{goldbring2015,
  author  = {Goldbring, Isaac and Sinclair, Thomas},
  title   = {Games and elementary equivalence of {II}$_1$ factors},
  journal = {Pacific J. Math.},
  volume  = {278},
  number  = {1},
  pages   = {103--118},
  year    = {2015},
  doi     = {10.2140/pjm.2015.278.103},
}
@misc{peterson-u3,
  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.3; version of 15 May 2026},
}
@article{connes1975,
  author  = {Connes, Alain},
  title   = {A factor not anti-isomorphic to itself},
  journal = {Ann. of Math.},
  volume  = {101},
  number  = {3},
  pages   = {536--554},
  year    = {1975},
  doi     = {10.2307/1970940},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-00996, "A II₁ factor whose ultrapowers differ from those of its opposite", clopen.solutions.