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Math-2813 · mathematics · Analysis

Abstract Cuntz-semigroup criterion for Jiang–Su absorption

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every separable nuclear complex C*-algebra A, is $A\cong A\otimes Z$ equivalent to an abstract isomorphism $Cu(A)\cong Cu(A\otimes Z)$ in the category Cu, without requiring that isomorphism to arise from the first-factor inclusion or any star-homomorphism?

Short English statement

For every separable nuclear complex C*-algebra A, is $A\cong A\otimes Z$ equivalent to an abstract isomorphism $Cu(A)\cong Cu(A\otimes Z)$ in the category Cu, without requiring that isomorphism to arise from the first-factor inclusion or any star-homomorphism?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2813-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2813

Manuscript 1 (family 291, 2026-09-23): 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.

Abstract Cuntz-semigroup criterion for Jiang–Su absorption

OpenAI's claim

A separable nuclear C*-algebra that is simple and non-elementary is claimed to absorb Jiang--Su under extended-functional strict comparison, including nonunital and unbounded-trace cases; semigroup divisibility also yields a nonsimple criterion. 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 $\operatorname{Cu}(A)$ be the ordered Cuntz semigroup of positive elements of $A\otimes\mathcal K$, modulo mutual norm-approximation comparison. For separable simple nuclear non-elementary $A$, OpenAI claims $A\simeq A\otimes_{\min}\mathcal Z$ if, for every $x,y\in\operatorname{Cu}(A)$, inequalities $\varphi(x)\le\varphi(y)$ for all additive order/supremum-preserving functionals $\varphi:\operatorname{Cu}(A)\to[0,\infty]$, strict whenever $0<\varphi(y)<\infty$, imply $x\le y$. It further claims absorption for every nonzero separable nuclear $A$ whose Cuntz semigroup satisfies both \[ (l+1)x\le ly\ \Longrightarrow\ x\le y\quad(l\ge1),\qquad \forall x,\ N\ge1\ \exists u:\ Nu\le x\le(N+1)u. \] In that case a unital $\mathcal Z\to (A_\omega\cap A')/\operatorname{Ann}(A)$ is asserted for every free ultrafilter, where $A_\omega$ is the norm ultrapower and $\operatorname{Ann}(A)=\{z:zA=Az=0\}$. Finally it claims \[ A\simeq A\otimes_{\min}\mathcal Z\quad\Longleftrightarrow\quad \operatorname{Cu}(A)\simeq\operatorname{Cu}(A\otimes_{\min}\mathcal Z) \] for all separable nuclear $A$, allowing an abstract ordered-semigroup isomorphism rather than only the canonical first-factor map. Here $\mathcal Z$ is the Jiang--Su algebra, and non-elementary excludes compact-operator algebras.

Outline of the argument

Manuscript 1 (family 291, 2026-09-23): Trace transfer into small hereditary supports followed by spectral rounding establishes full Cuntz-semigroup divisibility. Pure-state compression produces norm-visible matrix cones, while localized nuclear approximations produce trace-visible ones. Comparison packs their defects into a central corner, constructing dimension-drop maps whose lifted intertwining relations give Jiang--Su absorption.

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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