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Math-2817 · mathematics · Analysis

Equivariant Jiang–Su absorption for amenable actions

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
If a countable amenable group acts on a separable simple unital infinite-dimensional nuclear stably finite Z-stable complex C*-algebra, is that action cocycle conjugate to its tensor product with the trivial action on Z, without an outerness, trace-action or UCT assumption?

Short English statement

If a countable amenable group acts on a separable simple unital infinite-dimensional nuclear stably finite Z-stable complex C*-algebra, is that action cocycle conjugate to its tensor product with the trivial action on Z, without an outerness, trace-action or UCT assumption?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2817-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2817

Manuscript 1 (family 291, 2026-10-05): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Equivariant Jiang–Su absorption for amenable actions

OpenAI's claim

Amenable countable-group actions on simple unital separable nuclear stably finite Jiang--Su-stable algebras are claimed to absorb the trivial Jiang--Su action, without fixing the tracial-state simplex. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $A$ be a complex $C^*$-algebra, nuclear and stably finite, as well as simple, separable, unital and infinite-dimensional with $A\simeq A\otimes_{\min}\mathcal Z$, and let $\alpha$ be any action of a countable discrete amenable $G$. Write $B=A\otimes_{\min}\mathcal Z$, $\beta_g=\alpha_g\otimes\mathrm{id}$. OpenAI claims a unital $*$-isomorphism $\Phi:A\to B$ and unitaries $u_g\in B$ with \[ u_e=1,\qquad u_{gh}=u_g\beta_g(u_h),\qquad \Phi(\alpha_g(a))=u_g\beta_g(\Phi(a))u_g^*. \] Thus the action absorbs the trivial Jiang--Su action up to cocycle conjugacy. No freeness assumption on $\alpha$ is stated.

Outline of the argument

Manuscript 1 (family 291, 2026-10-05): Commuting orbit algebras produce independent rows in the uniform tracial central sequences. A covariance-sensitive Gaussian construction, normalized using square roots of Gram operators for Folner sets, gives estimates uniform over the trace space. Diagonal selections create an invariant central Haar unitary and equal-trace partitions, meeting the existing equivariant absorption criterion.

Formal verification

For family 291, the scope page documents the following selections. The selection gives uniform Gamma from real rank zero of the designated tracial ultrapower. The documented comparison scope also treats Jiang--Su absorption and Gamma under strict comparison, with additional trace and target-rank conditions in the projectionless case. Manuscript 1 is not among the papers selected by that page, so the family link does not formalize its main conclusion. It does not establish formal coverage of the complete problem clause.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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