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Tracial projection methods and uniform property Gamma

OpenAI · September 23, 2026

Lean

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Abstract

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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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). Describes the release's Lean formalization of Theorem 1.1 (real rank zero of the specified completion ultrapower implies uniform property $\Gamma$), and says linked comparison results also cover the paper's simple unital and bounded-trace stably projectionless absorption theorems.
  • OpenAI math release, UniformGamma.lean (comparator statement) (openai/math, commit adc7f12). The comparator statement CurrentMain.main : MainClaim (proof sorry in this file) states the Theorem 1.1 halving projection, existentially quantified over five construction typeclasses (norm, null, Cauchy, limit, trace) for the completion and ultrapower.
  • OpenAI math release, TracialSplitting/Challenge.lean (openai/math, commit adc7f12). Proves main from real_rank_zero_uniform_gamma via main_iff_original, which relates the comparator claim to the original uniform-$\Gamma$ statement.

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). Proves that strict comparison by traces implies pureness for unital simple non-elementary algebras (Corollary B), using trace-preserving compression and rank-rounding lemmas; this is a step toward the unital comparison-to-absorption theorem (u:thm:main) by a route that does not go through uniform property $\Gamma$, and it 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 of simple separable pure nuclear algebras using real rank zero, projection classification and hyperfiniteness of tracially complete algebras; together with Thiel this gives a route to the Toms–Winter conjecture independent of u:thm:main, and it relates to the paper's real-rank-zero theme (Theorem 1.1).

Digests and expositions

Elsewhere

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Cite

@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}
}

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: