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The Kirchberg-Rordam character question: does absence of characters on the central sequence algebra characterize Jiang-Su stability?

For a nonzero unital separable C*-algebra AA and a free ultrafilter ω\omega, let Fω(A)=Aω∩A′F_\omega(A)=A_\omega\cap A' be the norm central-sequence algebra. Kirchberg and Rordam proved A≅A⊗ZA\cong A\otimes\mathcal Z exactly when Fω(A)F_\omega(A) contains, with the same unit, a separable subhomogeneous algebra without characters; absence of characters on Fω(A)F_\omega(A) is then necessary. They asked (Question 3.1) whether it is sufficient, and related it to tensor powers (Question 3.7); the question reappears as Problem LXXX of Schafhauser, Tikuisis and White. Is AA Jiang-Su stable whenever Fω(A)F_\omega(A) has no characters?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Operator algebras; Jiang-Su stability and central sequences
Posed by
Eberhard Kirchberg and Mikael Rordam (When central sequence C*-algebras have characters, Internat. J. Math. 26 (2015), Questions 3.1 and 3.7); listed as Problem LXXX by Schafhauser, Tikuisis and White
Year posed
2014
Years open
12y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
20 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every nonzero unital separable complex C*-algebra AA and every free ultrafilter ω\omega, Fω(A)F_\omega(A) has no characters iff A≅A⊗min⁡ZA\cong A\otimes_{\min}\mathcal Z; no nuclearity, simplicity, trace or comparison hypothesis. Theorem 1.2: every nonzero unital characterless DD admits a unital map I(2,3)→D⊗max⁡mI(2,3)\to D^{\otimes_{\max}m} for some mm. Corollary: the infinite minimal tensor power of a nonzero unital separable characterless algebra is Z\mathcal Z-stable (bearing on Kirchberg-Rordam Question 3.7, via Dadarlat-Toms). Not shown: any finite tensor power is Z\mathcal Z-stable, or nonseparable cases.

What the AI did

Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.

Verification

No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1, read against Question 3.1 as the paper quotes it; proof not refereed. Lean-checked on Comparator challenge CharacterCriterion, declaration OAI.KirchbergRordam.character_criterion in OAI/Analysis/CharacterCriterion/Main.lean, listed in lean/formalization.yaml; statement read here, not rebuilt. The challenge file builds the norm ultrapower, central sequence algebra, minimal tensor product and the Jiang-Su algebra itself (about 2,100 lines of definitions), and states: for every nontrivial separable unital C*-algebra and free ultrafilter, no characters on the central algebra iff A≅A⊗min⁡ZA\cong A\otimes_{\min}\mathcal Z. That is the headline. The infinite tensor power corollary is not in the formal statement. The proof uses a companion OpenAI lemma (catalog entry robert-tikuisis-conjecture-c1).

Sources

Changelog1 change

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