VibeMathedMath problems solved with AI
Sign in

The generator problem: is every von Neumann algebra with separable predual singly generated?

The generator problem asks whether every von Neumann algebra MM with separable predual is W∗(x)W^*(x) for a single bounded operator xx, equivalently generated by two self-adjoint operators. Type I algebras (Pearcy), hyperfinite ones (Suzuki and Saito) and properly infinite ones (Wogen) were settled, and Willig's direct-integral theorem reduces the question to type II1\mathrm{II}_1 factors. Positive classes include factors with property Γ\Gamma or Cartan subalgebras (Popa, Ge-Popa), but the free group factors L(Fn)L(\mathbb F_n) were the outstanding unknown case. Is every von Neumann algebra with separable predual singly generated?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Operator algebras
Posed by
Classical problem; the manuscript names no poser, citing Pearcy (1962) for the first case and Sherman, Canad. J. Math. 64 (2012), Section 3, for its history
Year posed
—
Years open
—
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
50 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.2: every type II1\mathrm{II}_1 factor with separable predual is W∗(x)W^*(x) for one xx (two self-adjoint generators). Theorem 1.1: for an irreducible inclusion P⊂MP\subset M of II1\mathrm{II}_1 factors, MM separable, the unitaries uu with W∗(P,u)=MW^*(P,u)=M are a dense GδG_\delta in the 2-norm topology. With Willig's reduction this gives single generation of every von Neumann algebra with separable predual. Consequences: the generator invariants GG and GsaG_{sa} vanish, and Voiculescu's δ,δ0\delta,\delta_0 are not W∗W^*-invariants on L(Fn)L(\mathbb F_n), n≥3n\ge3. No explicit generator is constructed.

What the AI did

The release README says the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result, out of roughly 4,000 problems posed; the output was aggregated into result families and manuscripts and kept if judged significant enough. This family has one manuscript, dated September 23, 2026. The manuscript is credited to 'OpenAI' alone and names no human author. The README's two exceptions to the fixed procedure (the Riemann zeta zero-free region work, whose Re(s) > 11/12 write-up was human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The README also cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 and Section 5 of the TeX source read against the problem as stated. Theorem 1.2 proves the type II1\mathrm{II}_1 factor case; the full answer for all von Neumann algebras with separable predual follows only through Willig's published direct-integral reduction and the earlier type I and properly infinite cases, which the paper cites and does not reprove. The proof was not refereed. Lean: FactorGeneration (OAI.Generator.single_generation_of_II1_separable_predual) is a main result in the formalization catalogue. Its statement was read here: a WOT-closed unital star-subalgebra of B(H) that is a factor, finite and without minimal projections, with separable predual, equals the von Neumann algebra generated by one element. That is Theorem 1.2, not the reduction to general algebras. Not rebuilt here.

Sources

Changelog1 change

Discussion