Math Conjectures

Free-group-factor isomorphism problem

OPENLandmarkOpen problemProposed c. 1943 · Standard version

Canonical statement

For each integer n≥2n\ge2, let Fn\mathbb F_n be the free group on nn generators, let λn\lambda_n be its left regular representation on ℓ2(Fn)\ell^2(\mathbb F_n), and define the free group factor L(Fn)={λn(g):g∈Fn}′′L(\mathbb F_n)=\{\lambda_n(g):g\in\mathbb F_n\}''. Determine which of the following alternatives holds: L(Fm)≅L(Fn)L(\mathbb F_m)\cong L(\mathbb F_n) for every m,n≥2m,n\ge2, or L(Fm)≇L(Fn)L(\mathbb F_m)\not\cong L(\mathbb F_n) whenever m≠nm\ne n, where ≅\cong denotes a unital normal ∗*-isomorphism of von Neumann algebras.
View source LaTeX
For each integer \(n\ge2\), let \(\mathbb F_n\) be the free group on \(n\) generators, let \(\lambda_n\) be its left regular representation on \(\ell^2(\mathbb F_n)\), and define the free group factor \(L(\mathbb F_n)=\{\lambda_n(g):g\in\mathbb F_n\}''\). Determine which of the following alternatives holds: \(L(\mathbb F_m)\cong L(\mathbb F_n)\) for every \(m,n\ge2\), or \(L(\mathbb F_m)\not\cong L(\mathbb F_n)\) whenever \(m\ne n\), where \(\cong\) denotes a unital normal \(*\)-isomorphism of von Neumann algebras.
ANAL-FA-018:[Dykema1994InterpolatedFreeGroupFactors][Radulescu1994RandomMatricesFreeFactors][GoldbringPi2025FirstOrderFreeGroupFactors]
Analysis and functional analysisAlgebra and representation theory

Notes

The interpolation and amplification formulas of Dykema and Rădulescu prove that the two displayed alternatives exhaust the possibilities. No known von Neumann-algebra invariant distinguishes the rank, and no isomorphism between two distinct finite ranks is known.
This is the isomorphism problem for the tracial von Neumann algebras, not for the reduced group C∗C^*-algebras, whose ranks are distinguished by KK-theory. Year note: the approximate date follows the introduction of free group factors; no single first printed formulation of the rank problem was identified.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.