Browse › Operator algebras › 289 Strong Kadison–Kastler stability and its spatial boundaries
Universal strong Kadison–Kastler stability
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
We prove that sufficiently close unital von Neumann algebras are conjugate by a unitary arbitrarily close to the identity. The tolerance depends only on the prescribed distance of that unitary from the identity, uniformly over all algebras, representations, and Hilbert spaces. This resolves the strong Kadison–Kastler conjecture.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope note 289: Strong Kadison–Kastler stability and its spatial boundaries (lean/docs/289.md, 2026-10-06). Reports a formalization of the uniform strong Kadison–Kastler statement of Theorem 1.1 (near-identity unitary conjugacy for close unital von Neumann algebras, uniform over algebras, representations and Hilbert spaces); it lists only this paper of the family.
- OpenAI math release, StrongKadisonKastler.lean (comparator statement) (lean/ComparatorChallenges, 2026). The comparator file states
universal_strong_stability : UniversalStrongStabilitywith asorryplaceholder and is linked from the scope note as the formal counterpart of Theorem 1.1.
Possibly relevant literature (found by the agent, not checked)
- G. Pisier, The similarity degree of an operator algebra (arXiv:math/9706211; St. Petersburg Math. J. 10 (1999)). Similarity-degree theory relates to the paper's representation-uniform commutator (inner derivation) amplification bound for II$_1$ factors and its stated link with Kadison's similarity problem.
- L. Dickson, A Kadison Kastler row metric and intermediate subalgebras (arXiv:1404.3070, 2014; Internat. J. Math.). It compares a row version of the Kadison–Kastler metric with the ordinary one, which is relevant to the paper's distinction between ordinary distance and matrix-level gaps.
- F. Perera, A. Toms, S. White, W. Winter, The Cuntz semigroup and stability of close C*-algebras (arXiv:1210.4533; Anal. PDE 7 (2014)). It shows that close algebras with a positive answer to Kadison's similarity problem are completely close, which relates to the paper's passage from ordinary closeness to control at all matrix levels.
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:Universal-strong-Kadison-Kastler-stability-September-23-2026,
author = {{OpenAI}},
title = {{Universal strong Kadison--Kastler stability}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026/paper.pdf}{OAI:Universal-strong-Kadison-Kastler-stability-September-23-2026}},
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:
- Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings · October 5, 2026
- Close Separable C*-Algebras Without Spatial Conjugacy · October 5, 2026