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Small spatial embeddings for one-sided near inclusions of von Neumann algebras

For von Neumann algebras M,N⊆B(H)M,N\subseteq B(H) with common identity let γ(M,N)=sup⁡x∈M,∥x∥≤1inf⁡y∈N∥x−y∥\gamma(M,N)=\sup_{x\in M,\|x\|\le1}\inf_{y\in N}\|x-y\|. Christensen showed that an injective MM with small γ(M,N)\gamma(M,N) embeds into NN by a unitary close to the identity. Cameron, Christensen, Sinclair, Smith, White and Wiggins asked the unrestricted one-sided question: for every ε>0\varepsilon>0 is there a δ>0\delta>0 such that γ(M,N)<δ\gamma(M,N)<\delta always implies uMu∗⊆NuMu^*\subseteq N for some unitary uu with ∥u−1∥<ε\|u-1\|<\varepsilon?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Operator algebras; near inclusions, perturbation theory
Posed by
J. Cameron, E. Christensen, A. M. Sinclair, R. R. Smith, S. White and A. Wiggins, Type II_1 factors satisfying the spatial isomorphism conjecture, PNAS 109 (2012)
Year posed
2012
Years open
14y
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
20 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there are ε0>0\varepsilon_0>0 and pairs of von Neumann algebras on separable Hilbert spaces with common identity and γ(Mn,Nn)→0\gamma(M_n,N_n)\to0 for which spatial embeddings exist but every implementing unitary satisfies ∥u−1∥≥ε0\|u-1\|\ge\varepsilon_0. So the unrestricted one-sided question has a negative answer, in contrast with Christensen's injective case and with the two-sided strong Kadison-Kastler theorem in the same family. The obstruction is to small unitaries, not to the existence of embeddings.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the question as the paper quotes it (paragraph before Theorem 2 of Cameron et al. 2012); it claims algebras Mn,NnM_n,N_n on separable spaces with γ(Mn,Nn)→0\gamma(M_n,N_n)\to0 such that every unitary with uMnu∗⊆NnuM_nu^*\subseteq N_n stays at distance at least ε0\varepsilon_0 from the identity. Proof not refereed; not formalized.

Sources

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