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Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings
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Abstract
We give a negative answer to the unrestricted small-spatial-embedding problem for one-sided near inclusions of von Neumann algebras. On separable complex Hilbert spaces, we construct pairs of unital von Neumann algebras with common identity whose one-sided gaps tend to zero. Spatial embeddings of the source into the target exist, but every implementing unitary stays a fixed positive distance from the identity. The obstruction therefore concerns small implementing unitaries, rather than the existence of spatial embeddings.
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Possibly relevant literature (found by the agent, not checked)
- E. Christensen, A. M. Sinclair, R. R. Smith, S. White, C*-algebras nearly contained in type I algebras (arXiv:1107.1983; Canad. J. Math. 65 (2013)). It gives positive one-sided near-inclusion embedding results for a separable type I target when the source has the similarity property, a different setting from the amenable-source theorem the paper contrasts with its counterexample.
- J. Roydor, Near inclusions of amenable operator algebras (arXiv:1103.5225; Bull. London Math. Soc., 2011). It embeds amenable operator algebras nearly contained in complemented dual operator algebras via a similarity, a related one-sided positive result.
Digests and expositions
- Latent Seminar, Kadison–Kastler stability and its spatial boundaries (expository article, 2026-10-07). Explains all three manuscripts through positive-map correction, the almost-commuting-matrix obstruction, and sequence-algebra spatial rigidity; the article states that it reviews selected proof mechanisms rather than verifying the complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 289
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings-October-5-2026,
author = {{OpenAI}},
title = {{Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings-October-5-2026/near-inclusions.pdf}{OAI:Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings-October-5-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/289.md. The Lean statements have not been checked by people against the paper's claims.
Family 289: Strong Kadison–Kastler stability and its spatial boundaries
Proves that sufficiently close unital von Neumann algebras on the same Hilbert space are conjugate by a unitary arbitrarily close to the identity, with a universal tolerance in the operator-norm distance between unit balls. Counterexamples show that near-identity conjugacy fails for one-sided near inclusions, and that arbitrarily close norm-separable C∗-algebras need not be ambiently unitarily conjugate.
Other papers in this family:
- Universal strong Kadison–Kastler stability · September 23, 2026
- Close Separable C*-Algebras Without Spatial Conjugacy · October 5, 2026