OP-01050 Solution claimed

The free group factor problem

For $n \ge 2$, $L(\mathbb F_n)$ is the group von Neumann algebra of the free group on $n$ generators, a II$_1$ factor.

Problem. Is $L(\mathbb F_2)$ isomorphic to $L(\mathbb F_3)$?

Source: F. J. Murray and J. von Neumann, On rings of operators IV (1943); listed as O.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
Keywords: free group factors, II₁ factors, isomorphism problem, Peterson's operator algebra problems

References

@article{murray1943,
  author  = {Murray, F. J. and von Neumann, J.},
  title   = {On rings of operators. {IV}},
  journal = {Ann. of Math.},
  volume  = {44},
  number  = {4},
  pages   = {716--808},
  year    = {1943},
  doi     = {10.2307/1969107},
}
@misc{peterson-o2,
  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.2; version of 15 May 2026},
}
@article{dykema1994,
  author  = {Dykema, Kenneth},
  title   = {Interpolated free group factors},
  journal = {Pacific J. Math.},
  volume  = {163},
  number  = {1},
  pages   = {123--135},
  year    = {1994},
  doi     = {10.2140/pjm.1994.163.123},
}
@article{radulescu1994,
  author  = {R{\u{a}}dulescu, Florin},
  title   = {Random matrices, amalgamated free products and subfactors of the von {N}eumann algebra of a free group, of noninteger index},
  journal = {Invent. Math.},
  volume  = {115},
  number  = {1},
  pages   = {347--389},
  year    = {1994},
  doi     = {10.1007/BF01231764},
}

Resolution claims

An isomorphism of the free group factors v1 GitHub ↗

OpenAI

We solve the free group factor isomorphism problem affirmatively by proving that $L(\F_2)$ and $L(\F_3)$ are isomorphic as tracial von Neumann algebras. The classical free group factor alternative then implies that all interpolated free group factors, including $L(\F_\infty)$, are isomorphic and have fundamental group $\R_{>0}$.

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-29	Needs verification → Solution claimed	—	Recorded here: math @ 7ef5c0c, build/main.tex.

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-01050, "The free group factor problem", clopen.solutions.