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Close Separable C*-Algebras Without Spatial Conjugacy

OpenAI · October 5, 2026

Lean

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Abstract

We disprove the separable C∗-algebraic spatial form of the Kadison–Kastler conjecture. For every ε > 0, we construct unital, norm-separable C∗-algebras on a common separable complex Hilbert space, with the same identity and Kadison–Kastler distance less than ε, that are not conjugate by any unitary. The two algebras have the same von Neumann closure.

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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.

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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]

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Cite

@misc{OAI:Close-Separable-Cstar-Algebras-Without-Spatial-Conjugacy-October-5-2026,
  author = {{OpenAI}},
  title = {{Close Separable $C^*$-Algebras Without Spatial Conjugacy}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Close-Separable-Cstar-Algebras-Without-Spatial-Conjugacy-October-5-2026/paper.pdf}{OAI:Close-Separable-Cstar-Algebras-Without-Spatial-Conjugacy-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.

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