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Nuclear dimension and Jiang–Su stability without elementary subquotients

OpenAI · September 23, 2026

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Abstract

For separable nuclear C∗-algebras with no nonzero elementary ideal subquotients, finite nuclear dimension is equivalent to Jiang–Su stability. More generally, $\dim_{\mathrm{nuc}}(A_0\otimes\mathcal Z)\le1$ for every separable nuclear A0. This proves Robert and Tikuisis's Conjecture (C1) and the nuclear-dimension equivalence in the nonsimple Toms–Winter regularity question.

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

  • S. Evington, S. Hua, C. Schafhauser, A. Seth, A. Tikuisis, S. White, Regular simple nuclear C*-algebras (arXiv:2610.09390, 7 October 2026). Its Theorem A shows that for simple separable nuclear non-elementary algebras, finite nuclear dimension, nuclear dimension at most one, $\mathcal Z$-stability, almost unperforation and pureness are equivalent. This is the simple case of the paper's main theorem (thm:main) and its Cuntz-semigroup corollary (cor:regularity-cu), obtained independently and without citing the OpenAI manuscripts.

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Cite

@misc{OAI:Nuclear-dimension-and-Jiang-Su-stability-without-elementary-subquotients-September-23-2026,
  author = {{OpenAI}},
  title = {{Nuclear dimension and Jiang--Su stability without elementary subquotients}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Nuclear-dimension-and-Jiang-Su-stability-without-elementary-subquotients-September-23-2026/paper.pdf}{OAI:Nuclear-dimension-and-Jiang-Su-stability-without-elementary-subquotients-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.

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