Robert's trace-cone question: does the cone of extended traces classify W-stabilized separable nuclear C*-algebras?
The Razak-Jacelon algebra is simple, monotracial, stably projectionless and nuclear; tensoring with it removes K-theory and leaves trace data. For a C*-algebra let be the cone of all lower-semicontinuous extended traces , including the zero/infinity weights of ideals, with its canonical topology (Elliott-Robert-Santiago). Known classification results cover simple algebras and the traceless (purely infinite) case. Leonel Robert asked, as recorded by Santiago (2012) and listed as Problem LXVIII by Schafhauser, Tikuisis and White, whether this cone is a complete invariant: if are separable nuclear C*-algebras with as topological cones, is ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Operator algebras; classification of nuclear C*-algebras
- Posed by
- Leonel Robert, recorded in L. Santiago, W-stable C*-algebras (West Coast Operator Algebra Seminar abstract, 2012); Problem LXVIII in Schafhauser-Tikuisis-White, Nuclear C*-algebras: 99 problems
- Year posed
- 2012
- Years open
- 14y
- Solved
- 2026-09-25
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: if are separable nuclear complex C*-algebras and as topological cones, then , and the isomorphism can be chosen to induce the given cone map. No density condition on finite domains of the weights and no UCT is assumed. It does not classify algebras without the stabilization, and does not determine the range of the invariant.
What the AI did
The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Robert's question in the Schafhauser-Tikuisis-White formulation, which keeps every extended trace including the ideal weights. The theorem assumes only separability and nuclearity, allows arbitrary ideal structure and both finite and infinite subquotients, and needs no UCT. The release has no Lean formalization for this family (lean/docs/301.md does not exist at the pinned commit). The proof combines existence and uniqueness theorems with an approximate intertwining; it was not refereed here.