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Math-2385 · mathematics · Algebra

Nuclear Dimension after Jiang–Su Stabilization

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Is $\operatorname{dim}_{\mathrm{nuc}}(A\otimes\mathcal Z)<\infty$ for every nuclear $C^*$-algebra $A$? What is the decomposition rank of $C(X)\subset C(X,M_n)$? Is it less than $\dim X$, or does dimension drop only after tensoring with a UHF algebra?

Formulation references: [S1], [S2].

Short English statement

Is $\operatorname{dim}_{\mathrm{nuc}}(A\otimes\mathcal Z)<\infty$ for every nuclear $C^*$-algebra $A$? What is the decomposition rank of $C(X)\subset C(X,M_n)$? Is it less than $\dim X$, or does dimension drop only after tensoring with a UHF algebra?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2385-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2385

Manuscript 1 (family 291, 2026-09-23): The bound dim_nuc(A$\otimes$Z)$\leq$1 proves the first branch for separable nuclear A; the record also asks decomposition rank of the C(X) inclusion and permits nonseparable nuclear algebras.

Nuclear Dimension after Jiang–Su Stabilization

OpenAI's claim

Finite nuclear dimension is claimed to characterize Jiang--Su stability for separable nuclear algebras lacking elementary ideal subquotients; moreover, Jiang--Su stabilization always lowers nuclear dimension to at most one. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For a separable nuclear complex $C^*$-algebra $A$ with no ideal subquotient that is nonzero and elementary $I/J\simeq\mathcal K(H)$, OpenAI claims \[ \dim_{\mathrm{nuc}}A<\infty\quad\Longleftrightarrow\quad A\simeq A\otimes_{\min}\mathcal Z\quad\Longleftrightarrow\quad\dim_{\mathrm{nuc}}A\le1. \] Nuclear dimension is defined by finite-dimensional completely positive approximations with order-zero color decomposition; $\mathcal Z$ is the Jiang--Su algebra. The conditions are also claimed equivalent to either a canonical or an abstract isomorphism of $\operatorname{Cu}(A)$ with $\operatorname{Cu}(A\otimes_{\min}\mathcal Z)$. Without the subquotient restriction it asserts $\dim_{\mathrm{nuc}}(A_0\otimes_{\min}\mathcal Z)\le1$ for every separable nuclear $A_0$. Separately, for every nonzero unital characterless $D$ it claims a finite maximal tensor power containing two full orthogonal positive elements; full means generating the whole algebra as a closed ideal.

Outline of the argument

Manuscript 1 (family 291, 2026-09-23): Localized order-zero approximations control ranks even when traces are infinite on some ideals. A supporting map and comparison yield two-color approximations that survive stabilization and hereditary passage. Conversely, Gaussian sign tensors and pure-state excision construct two full orthogonal central elements from the absence of characters, supplying the 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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