OP-01053 Solution claimed

Are ℓ²-Betti numbers of groups invariants of their group von Neumann algebras?

Connes and Shlyakhtenko defined $\ell^2$-Betti numbers for tracial von Neumann algebras and showed that those of the group algebra $\mathbb C\Gamma$ are the $\ell^2$-Betti numbers $\beta_n^{(2)}(\Gamma)$ of the group.

Problem. Let $\Gamma$ and $\Lambda$ be countable groups with $L\Gamma \cong L\Lambda$. Must $\beta_n^{(2)}(\Gamma) = \beta_n^{(2)}(\Lambda)$ for every $n$?

Source: A. Connes and D. Shlyakhtenko, L²-homology for von Neumann algebras (2005); listed as I.1 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: group von Neumann algebras, ℓ²-Betti numbers, Peterson's operator algebra problems, W*-invariants

References

@article{connes2005,
  author  = {Connes, Alain and Shlyakhtenko, Dimitri},
  title   = {{$L^2$}-homology for von {N}eumann algebras},
  journal = {J. Reine Angew. Math.},
  volume  = {586},
  pages   = {125--168},
  year    = {2005},
  doi     = {10.1515/crll.2005.2005.586.125},
}
@misc{peterson-i1,
  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 I.1; version of 15 May 2026},
}
@article{popavaes2015,
  author        = {Popa, Sorin and Vaes, Stefaan},
  title         = {Vanishing of the first continuous {$L^2$}-cohomology for {II}$_1$ factors},
  journal       = {Int. Math. Res. Not. IMRN},
  year          = {2014},
  doi           = {10.1093/imrn/rnu054},
  eprint        = {1401.1186},
  archivePrefix = {arXiv},
}
@article{shlyakhtenko2021,
  author  = {Shlyakhtenko, Dimitri},
  title   = {Von {N}eumann algebras of sofic groups with {$\beta_1^{(2)} = 0$} are strongly 1-bounded},
  journal = {J. Operator Theory},
  volume  = {85},
  number  = {1},
  pages   = {216--228},
  year    = {2021},
  doi     = {10.7900/jot.2019oct21.2270},
}

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 $\R_{>0}$.

Counterexample · 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-01053, "Are ℓ²-Betti numbers of groups invariants of their group von Neumann algebras?", clopen.solutions.