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Math-2827 · mathematics · Analysis

Small spatial embeddings from one-way near-inclusions

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Does arbitrarily small one-way unit-ball distance from a unital von Neumann algebra M to N, acting on the same Hilbert space, force a unitary u arbitrarily close to 1 with $uMu*\subseteq N$? The claim concerns small implementing unitaries, not the existence of an unrestricted spatial embedding.

Short English statement

Does arbitrarily small one-way unit-ball distance from a unital von Neumann algebra M to N, acting on the same Hilbert space, force a unitary u arbitrarily close to 1 with $uMu*\subseteq N$? The claim concerns small implementing unitaries, not the existence of an unrestricted spatial embedding.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2827-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2827

Manuscript 1 (family 289, 2026-10-05): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed counterexample to that assertion in full.

Small spatial embeddings from one-way near-inclusions

OpenAI's claim

On separable Hilbert spaces, one-sided near inclusions can have arbitrarily small gaps while every spatial embedding requires a unitary bounded away from the identity, despite embeddings existing. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Define the one-sided near-inclusion distance $\gamma(M,N)=\sup_{x\in M_1}\inf_{y\in N}\|x-y\|$. OpenAI claims $\varepsilon_0>0$, Hilbert spaces $H_n$ that are separable, together with pairs of von Neumann algebras $M_n,N_n\subset B(H_n)$ sharing the identity, for which \[ \gamma(M_n,N_n)\to0, \qquad uM_nu^*\subseteq N_n\ \Longrightarrow\ \|u-I_{H_n}\|\ge\varepsilon_0 \] for every unitary $u$ and every sufficiently large $n$. This rules out a universal small-unitary embedding bound from one-sided closeness; it does not state nonexistence of all spatial embeddings.

Outline of the argument

Manuscript 1 (family 289, 2026-10-05): The construction inserts Voiculescu clock-and-shift matrices into a circle-parameter deformation of a free-group representation. Summable Fourier estimates make the resulting algebras nearly included. Weak limits along distinct free generators retain the original matrices, so a hypothetical small implementing unitary would violate their noncommuting winding-number obstruction.

Formal verification

For family 289, the scope page documents the following selections. The selection gives a tolerance depending only on the requested distance of an implementing unitary from identity, for two close common-identity von Neumann algebras. The separate one-way near-inclusion and general norm-closed examples are outside it. 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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