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The Toms-Winter conjecture: strict comparison implies Jiang-Su stability for simple nuclear C*-algebras

For simple separable unital infinite-dimensional nuclear C*-algebras AA, the Toms-Winter conjecture asserts the equivalence of three regularity properties: finite nuclear dimension, Jiang-Su stability A≅A⊗ZA\cong A\otimes\mathcal Z, and strict comparison of positive elements by traces. Winter and Castillejos-Evington-Tikuisis-White-Winter proved finite nuclear dimension equivalent to Z\mathcal Z-stability, and Rordam proved that Z\mathcal Z-stability gives strict comparison. The remaining implication, strict comparison implies Z\mathcal Z-stability, was known only under conditions on the tracial boundary (finitely many extreme traces, compact finite-dimensional boundary, Lin's condition) or with uniform property Γ\Gamma. Does strict comparison imply Z\mathcal Z-stability for every simple separable nuclear non-elementary C*-algebra?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Operator algebras; regularity in the Elliott classification program
Posed by
Andrew Toms and Wilhelm Winter
Year posed
2010
Years open
16y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
48 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: strict comparison implies A≅A⊗ZA\cong A\otimes\mathcal Z for every separable simple nuclear non-elementary AA, unital or not. More generally (Theorem 1.2) every separable nuclear algebra whose Cuntz semigroup is almost unperforated and fully almost divisible is Z\mathcal Z-stable, which answers the implication (iii) to (ii) of the nonsimple Toms-Winter problem (STW Problem LXXV) and the abstract Cuntz-semigroup question (STW Problem XXVI). Separability and nuclearity remain hypotheses; nonnuclear and non-separable algebras are not treated.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's 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. The family has four manuscripts. This entry's principal is 'Cuntz comparison and Jiang-Su absorption' (September 23, 2026); 'Tracial projection methods and uniform property Gamma' (same date) gives an independent proof in the unital case.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against the conjecture. It states that every separable simple nuclear non-elementary C*-algebra with strict comparison (in the extended-functional convention, which it shows equals almost unperforation of the Cuntz semigroup) is Z\mathcal Z-stable, including nonunital algebras with unbounded traces. With the known equivalence of finite nuclear dimension and Z\mathcal Z-stability this completes the conjecture. The tracial-projection companion proves the unital case independently. Neither proof was refereed. The family's Lean comparator (UniformGamma) states only the companion's Problem XXI result, not this implication, so the entry is unreviewed.

Sources

Changelog1 change

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