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Math-2383 · mathematics · Algebra

Subhomogeneous Character-Free Subalgebras of Infinite Tensor Powers

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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.
Let $B$ be a separable unital $C^*$-algebra without characters and with separating tracial states. Does $B\otimes_{\min}B\otimes_{\min}\cdots$ contain a subhomogeneous $C^*$-algebra without characters?

Formulation references: [S1], [S2].

Short English statement

Let $B$ be a separable unital $C^*$-algebra without characters and with separating tracial states. Does $B\otimes_{\min}B\otimes_{\min}\cdots$ contain a subhomogeneous $C^*$-algebra without characters?

Sources

Catalogue entry

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

Contributions 1

Math-2383-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2383

Manuscript 1 (family 299, 2026-09-25): The infinite minimal tensor power of any unital separable character-free B contains Z, hence a unital prime dimension-drop subalgebra without characters; no separating-traces assumption is needed.

Subhomogeneous Character-Free Subalgebras of Infinite Tensor Powers

OpenAI's claim

For nonzero unital separable C*-algebras, absence of characters in the norm central sequences is claimed to characterize Jiang--Su absorption; characterless algebras also have Jiang--Su-stable infinite minimal tensor powers. 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 nonzero unital separable complex $C^*$-algebra $A$ and free ultrafilter $\omega$, define its norm ultrapower $A_\omega$ and central-sequence algebra $F_\omega(A)=A_\omega\cap A'$, using the diagonal copy of $A$. OpenAI claims \[ F_\omega(A)\text{ has no nonzero }*\text{-homomorphism to }\mathbb C \quad\Longleftrightarrow\quad A\simeq A\otimes_{\min}\mathcal Z, \] where $\mathcal Z$ is the Jiang--Su algebra. It additionally claims that for every nonzero unital separable characterless $D$, its infinite minimal tensor power $D^{\otimes_{\min}\infty}=\varinjlim(D^{\otimes n},x\mapsto x\otimes1)$ absorbs $\mathcal Z$ tensorially and admits a unit-preserving inclusion of $\mathcal Z$.

Outline of the argument

Manuscript 1 (family 299, 2026-09-25): Finite tensor labels detect noncommutativity in every representation and build a full square-zero element. Its two orthogonal corners are exactly equivalent, and a finite comparison absorbs their unit defect into a positive cutoff. This supplies dimension-drop maps; reindexed commuting copies then pass from the finite tensor construction to central-sequence absorption and infinite tensor powers.

Formal verification

For family 299, the scope page documents the following selections. The selection equates Jiang--Su absorption with absence of characters in the norm central-sequence algebra for nonzero unital separable algebras, without nuclearity or trace assumptions. The tensor-power consequence is separate. The documented statement covers the main claim of manuscript 1. It does not establish formal coverage of the complete problem clause. Comparator files: CharacterCriterion.lean. Declarations: OAI.KirchbergRordam.character_criterion.

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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