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Equivariant Jiang–Su Stability for Amenable Actions in the Unital Stably Finite Case
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Abstract
Every action of a countable discrete amenable group on a simple, separable, unital, infinite-dimensional, nuclear, stably finite complex C∗-algebra that is already Jiang–Su stable absorbs the trivial action on the Jiang–Su algebra up to cocycle conjugacy. No restriction is imposed on the action on the tracial-state simplex. This proves the unital, stably finite case of Szabó's Conjecture A on automatic equivariant Jiang–Su stability.
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Related prior literature mentioned by others
- Gábor Szabó, Equivariant property (SI) revisited (Analysis & PDE 14, 2021). Named on VibeMathed as the source of the conjecture; it proves equivariant Jiang–Su stability under strict comparison and finitely many extremal trace rays.
Possibly relevant literature (found by the agent, not checked)
- E. Gardella, I. Hirshberg, A. Vaccaro, Strongly outer actions of amenable groups on $\mathcal Z$-stable nuclear $C^*$-algebras (arXiv:2110.14387, 2021). An earlier positive case of automatic equivariant $\mathcal Z$-stability for amenable actions, assuming bounded finite trace orbits and a compact finite-dimensional extreme boundary with Hausdorff orbit space; relevant to the paper's claim that Theorem 1.1 needs no condition on trace orbits.
- L. Wouters, Equivariant $\mathcal Z$-stability for single automorphisms on simple $C^*$-algebras with tractable trace simplices (arXiv:2105.04469, 2021; v3 2023, to appear in Math. Z. per arXiv comments). Proves cocycle conjugacy $\alpha\simeq\alpha\otimes\mathrm{id}_{\mathcal Z}$ for every automorphism in the Bauer case with finite-dimensional boundary, a special case of Theorem 1.1 (with $G=\mathbb Z$) that the paper does not cite.
Digests and expositions
- Latent Seminar, Jiang–Su stability: comparison, dimension and central splitting (7 October 2026). Explains the hypotheses of the four manuscripts and some of their proof mechanisms, and says explicitly that it has not independently verified the complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 291
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026,
author = {{OpenAI}},
title = {{Equivariant Jiang--Su Stability for Amenable Actions in the Unital Stably Finite Case}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/paper.pdf}{OAI:Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026}},
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:
- Cuntz comparison and Jiang–Su absorption · September 23, 2026
- Nuclear dimension and Jiang–Su stability without elementary subquotients · September 23, 2026
- Tracial projection methods and uniform property Gamma · September 23, 2026