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Math-2286 · mathematics · Analysis

An ultrafilter-free local criterion for O2-ultrapower embeddability

Status Partially solved · unsolvedmath.comLLM status Open unverified

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.
Does every separable C*-algebra embed into an ultrapower of $\mathcal O_2$ with respect to an ultrafilter on $\mathbb N$?

Formulation references: [S1], [S2].

Short English statement

Does every separable C*-algebra embed into an ultrapower of $\mathcal O_2$ with respect to an ultrafilter on $\mathbb N$?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2286-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2286

Manuscript 1 (family 292, 2026-09-23): The separable unital counterexample cannot embed unitally into $\mathcal O_2^\omega$; any nonunital embedding sends its unit to a nonzero projection, which is equivalent to 1 coordinatewise in O2, so passing to that corner would give a forbidden unital embedding.

An ultrafilter-free local criterion for O2-ultrapower embeddability

OpenAI's claim

One explicit separable unital full group C*-algebra is claimed to embed into no norm ultrapower of a nonzero unital nuclear algebra, giving a negative instance for O_2 as well. 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 \[ G=\mathbb Z[1/2]^3\rtimes(\operatorname{SL}_3(\mathbb Z)\times\mathbb Z),\qquad (M,k)v=2^kMv,\qquad A=C^*(G). \] OpenAI claims that for any nuclear $C^*$-algebra $B$ assumed nonzero and unital, and for any free ultrafilter $\omega$ on $\mathbb N$, a unital embedding $A\hookrightarrow B^\omega$ is impossible. This also excludes $\mathcal O_2^\omega$ as a target. The algebra $A$ is the full group algebra and is separable and unital; $B^\omega=\ell^\infty(\mathbb N,B)/\{(b_n):\lim_\omega\|b_n\|=0\}$ is the norm ultrapower. This addresses nuclear norm-ultrapower embeddings.

Outline of the argument

Manuscript 1 (family 292, 2026-09-23): An ascending HNN construction supplies a property-T base with finite-index self-embeddings. A nuclear tensor-product estimate restricts the finite-dimensional representation types approximated by arbitrary unitary tuples, uniformly in dimension. Applying this to coordinate lifts of a proposed ultrapower embedding and then inducing through the self-embedding produces incompatible finite sets of representation types.

Formal verification

For family 292, the scope page documents the following selections. The selected full-group algebra has no unital embedding in a norm ultrapower of any nonzero unital nuclear target. In particular, the Cuntz-algebra ultrapower target is excluded. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. Comparator files: NuclearUltrapower.lean. Declarations: OAI.NuclearUltrapower.main_no_embedding.

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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