Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability

4 papersLean

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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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 4

Equivariant Jiang–Su Stability for Amenable Actions in the Unital Stably Finite Case

OpenAI · October 5, 2026

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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@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}
}

Cuntz comparison and Jiang–Su absorption

OpenAI · September 23, 2026

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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@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}
}

Nuclear dimension and Jiang–Su stability without elementary subquotients

OpenAI · September 23, 2026

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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@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}
}

Tracial projection methods and uniform property Gamma

OpenAI · September 23, 2026

For a simple, separable, unital, infinite-dimensional, nuclear, stably finite C^∗-algebra with traces, real rank zero of the uniform tracial ultrapower of its uniform tracial completion implies uniform property Γ. This answers Problem XXI of Schafhauser, Tikuisis and White affirmatively. Independently, strict comparison implies Jiang–Su absorption for simple, separable, unital, infinite-dimensional nuclear algebras, resolving the unital Toms–Winter conjecture. Under comparison tested on traces of finite target rank, we also obtain uniform property Γ and absorption for simple, separable, nuclear, stably projectionless algebras whose densely finite traces are all bounded and have a nonempty compact normalized base.

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@misc{OAI:Tracial-projection-methods-and-uniform-property-Gamma-September-23-2026,
  author = {{OpenAI}},
  title = {{Tracial projection methods and uniform property $\Gamma$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Tracial-projection-methods-and-uniform-property-Gamma-September-23-2026/paper.pdf}{OAI:Tracial-projection-methods-and-uniform-property-Gamma-September-23-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Tracial projection methods and uniform property \(\Gamma\)

Scope

The formalization proves that real rank zero of the uniform tracial ultrapower of the uniform tracial completion implies uniform property \(\Gamma\) for a simple separable unital infinite-dimensional nuclear stably finite \(C^*\)-algebra with traces. The conclusion holds at every specified free ultrafilter under the corresponding real-rank-zero hypothesis.

The linked comparison results also prove Jiang–Su absorption and uniform property \(\Gamma\) for simple separable unital infinite-dimensional nuclear algebras with strict comparison, and for the stated stably projectionless nuclear algebras with bounded densely finite traces, a nonempty compact normalized trace base, and the prescribed finite-target-rank comparison condition.

Comparator links

Result	Comparator statement
Uniform property \(\Gamma\) from tracial real rank zero	UniformGamma.lean

Scope note on GitHub →