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Cuntz comparison and Jiang–Su absorption

OpenAI · September 23, 2026

Lean

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Abstract

We prove that strict comparison in the extended-functional sense implies Jiang–Su absorption for separable simple nuclear non-elementary C∗-algebras. This resolves the corresponding implication of the Toms–Winter regularity problem, including nonunital algebras and allowing unbounded traces. More generally, every separable nuclear C∗-algebra whose Cuntz semigroup is almost unperforated and fully almost divisible absorbs the Jiang–Su algebra.

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Formalization

  • OpenAI math release, Lean scope notes for family 291 (lean/docs/291.md, October 2026). According to the scope note, the formalized comparison-to-absorption results belong to the tracial-projection companion paper and cover the simple unital case and a bounded-trace stably projectionless case, which overlap parts of this paper's simple comparison theorem. [family-level]

Possibly relevant literature (found by the agent, not checked)

  • H. Thiel, Ranks of operators and pureness of C*-algebras (arXiv:2610.06761, 5 October 2026). Shows that for unital simple non-elementary algebras, strict comparison by traces gives almost divisibility and pureness without assuming exactness, using trace-preserving compression and rank-rounding lemmas; this overlaps with the paper's simple divisibility theorem (ext:thm:divisibility), and Thiel's paper does not cite the OpenAI papers.
  • S. Evington, S. Hua, C. Schafhauser, A. Seth, A. Tikuisis, S. White, Regular simple nuclear C*-algebras (arXiv:2610.09390, 7 October 2026). Proves $\mathcal Z$-stability for all simple separable pure nuclear algebras (Theorem A, no unit assumed), which together with Thiel's work gives an independent route to the comparison-to-$\mathcal Z$-stability implication in Theorem 1.1. It does not cite the OpenAI manuscripts and says one lemma was first obtained with ChatGPT Astra.
  • J. Castillejos, S. Evington, Stabilising uniform property Gamma (arXiv:2010.05792, 2020; Proc. Amer. Math. Soc.). Proves $\mathcal Z$-stability for simple separable nuclear algebras with stabilised property Gamma and $\mathrm{Cu}(A)\cong\mathrm{Cu}(A\otimes\mathcal Z)$, an earlier simple-case relative of the paper's Cuntz-semigroup criteria (cor:canonical, cor:abstract-cu) that the paper does not cite.

Digests and expositions

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Cuntz-comparison-and-Jiang-Su-absorption-September-23-2026,
  author = {{OpenAI}},
  title = {{Cuntz comparison and Jiang--Su absorption}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Cuntz-comparison-and-Jiang-Su-absorption-September-23-2026/paper.pdf}{OAI:Cuntz-comparison-and-Jiang-Su-absorption-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/291.md. The Lean statements have not been checked by people against the paper's claims.

Family 291: Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability

Proves equivariant Jiang–Su stability for every countable discrete amenable group action on a simple separable unital infinite-dimensional nuclear stably finite Jiang–Su-stable C∗-algebra, resolving this case of Szabó's conjecture without restrictions on trace dynamics. The family also proves the unital Toms–Winter conjecture, equating strict comparison, finite nuclear dimension and Jiang–Su stability in the simple separable unital infinite-dimensional nuclear setting.

Other papers in this family: