OP-01003 Open

Discrete generating copies of F₂ in unitary groups of II₁ factors

Problem. Let $M$ be a II$_1$ factor with trace $\tau$, and $\pi : \mathbb F_2 \to \mathcal U(M)$ an injective homomorphism whose image is discrete in the strong operator topology and generates $M$. Does $\pi$ extend to a $*$-isomorphism $L(\mathbb F_2) \to M$, that is, is $\tau(\pi(g)) = 0$ for every $g \ne e$?

Source: Credited to A. Thom (2011) and listed as H.2 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: discrete subgroups, free group factors, Peterson's operator algebra problems, unitary groups
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{peterson-h2,
  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 H.2; version of 15 May 2026},
}
@article{alekseev2022-h2,
  author  = {Alekseev, Vadim and Thom, Andreas},
  title   = {Maximal discrete subgroups in unitary groups of operator algebras},
  journal = {SIGMA},
  volume  = {18},
  pages   = {052},
  year    = {2022},
  doi     = {10.3842/SIGMA.2022.052},
}
@article{carderi2018-h2,
  author  = {Carderi, Alessandro and Thom, Andreas},
  title   = {An exotic group as limit of finite special linear groups},
  journal = {Ann. Inst. Fourier},
  volume  = {68},
  number  = {1},
  pages   = {257--273},
  year    = {2018},
  doi     = {10.5802/aif.3160},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-01003, "Discrete generating copies of F₂ in unitary groups of II₁ factors", clopen.solutions.