OP-00961 Open

Does every nonamenable II₁ factor contain the free group factor L(F₂)?

This is the analogue, for II$_1$ factors, of von Neumann's question whether every nonamenable group contains a free subgroup on two generators.

Problem. Let $M$ be a nonamenable II$_1$ factor. Must $M$ contain a von Neumann subalgebra isomorphic to $L(\mathbb F_2)$?

Source: Credited to S. Popa (1986), as the analogue for II₁ factors of von Neumann's problem for groups, and listed as O.8 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: GR
Keywords: amenability, free group factors, II₁ factors, Peterson's operator algebra problems
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{peterson-o8,
  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 O.8; version of 15 May 2026},
}
@article{berendschot2025,
  author  = {Berendschot, Tey and Vaes, Stefaan},
  title   = {Measure equivalence embeddings of free groups and free group factors},
  journal = {Ann. Sci. {\'E}c. Norm. Sup{\'e}r.},
  volume  = {58},
  pages   = {389--418},
  year    = {2025},
  doi     = {10.24033/asens.2607},
}
@article{gaboriau2009,
  author  = {Gaboriau, Damien and Lyons, Russell},
  title   = {A measurable-group-theoretic solution to von {N}eumann's problem},
  journal = {Invent. Math.},
  volume  = {177},
  number  = {3},
  pages   = {533--540},
  year    = {2009},
  doi     = {10.1007/s00222-009-0187-5},
}
@article{olshanskii1980,
  author  = {Ol'shanskii, A. Yu.},
  title   = {On the problem of the existence of an invariant mean on a group},
  journal = {Russian Math. Surveys},
  volume  = {35},
  number  = {4},
  pages   = {180--181},
  year    = {1980},
  doi     = {10.1070/RM1980v035n04ABEH001876},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-00961, "Does every nonamenable II₁ factor contain the free group factor L(F₂)?", clopen.solutions.