Result 289, Operator algebras

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.

Lean formalization Proof

The bigger picture

Why it matters

Two collections of bounded linear transformations can look almost identical without obviously being the same. These manuscripts claim a setting where closeness guarantees a small change of coordinates, and identify boundaries where that guarantee fails.

What changes?

For unital von Neumann algebras on the same Hilbert space, the stability manuscript reports a universal guarantee. These are algebras of bounded operators containing the identity and closed under adjoints and weak operator limits. Distance compares their unit balls in both directions using operator norm. Any prescribed positive bound on a conjugating unitary's distance from the identity has a sufficient tolerance depending only on that bound, uniformly over all algebras, representations and Hilbert spaces.

What does that help mathematicians do?

The reported counterexamples delimit what researchers can infer from closeness. On separable complex Hilbert spaces, unital von Neumann algebras sharing an identity can have one-sided gaps tending to zero and admit unitary embeddings, while every implementing unitary remains a fixed positive distance from the identity. Also, unital C*-algebras with countable norm-dense subsets can be arbitrarily close on a common separable complex Hilbert space, sharing an identity, without any unitary conjugacy. Thus neither extension follows from the positive result.

Are there practical applications?

The immediate value is foundational: the positive claim makes two-sided perturbations controllable through norm-preserving changes of coordinates. The C*-algebra examples, which concern norm-closed operator algebras, also share the same von Neumann closure. This shows that passing to that closure can erase distinctions that prevent unitary conjugacy of the original algebras.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

Universal strong Kadison–Kastler stability

September 23, 2026 70 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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}
}

Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings

October 5, 2026 10 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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}
}

Close Separable C*-Algebras Without Spatial Conjugacy

October 5, 2026 9 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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's note on what the formalization covers, from lean/docs/289.md.

Strong Kadison–Kastler stability and its spatial boundaries

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Universal strong Kadison–Kastler stability

Scope

The formalization proves the strong Kadison–Kastler conjecture uniformly. For every $\varepsilon>0$ , there is a $\delta>0$ such that any two unital von Neumann algebras on the same complex Hilbert space at Kadison–Kastler distance below $\delta$ are conjugate by a unitary $u$ with $\lVert u-1\rVert<\varepsilon$ . The tolerance depends only on $\varepsilon$ , uniformly over the algebras, their representations, and the Hilbert space.

Comparator links

Result	Comparator statement
Universal near-identity unitary conjugacy for close von Neumann algebras	StrongKadisonKastler.lean