Let H be a Hilbert space, let B( H) denote the bounded operators on H, and let d(M,N) be the Hausdorff distance between the unit balls of von Neumann algebras M,N⊆ B( H), measured in the operator norm. Kadison–Kastler conjecture. For all varepsilon>0, there exists delta>0 with the property that if M,N⊆ B( H) are von Neumann algebras with d(M,N)<delta, then there exists a unitary operator u on H with uMu^=N and |u-operatornameid_ H|<varepsilon. The conjecture asserts that sufficiently close von Neumann algebras are related by a small unitary perturbation. It was established for amenable von Neumann algebras, while the paper describes examples of non-amenable algebras satisfying it; the general status is not established by the supplied text.

Kadison–Kastler conjecture for von Neumann algebras

About 14 years old · traced to 2012

Let $\mathcal H$ be a Hilbert space, let $\mathcal B(\mathcal H)$ denote the bounded operators on $\mathcal H$ , and let $d(M,N)$ be the Hausdorff distance between the unit balls of von Neumann algebras $M,N\subseteq\mathcal B(\mathcal H)$ , measured in the operator norm. Kadison–Kastler conjecture. For all $\varepsilon>0$ , there exists $\delta>0$ with the property that if $M,N\subseteq\mathcal B(\mathcal H)$ are von Neumann algebras with $d(M,N)<\delta$ , then there exists a unitary operator $u$ on $\mathcal H$ with $uMu^*=N$ and $\|u-\operatorname{id}_{\mathcal H}\|<\varepsilon$ . The conjecture asserts that sufficiently close von Neumann algebras are related by a small unitary perturbation. It was established for amenable von Neumann algebras, while the paper describes examples of non-amenable algebras satisfying it; the general status is not established by the supplied text.

References

Primary source

Jan Cameron, Erik Christensen, Allan M. Sinclair, Roger R. Smith, Stuart White and Alan D. Wiggins, “Type II_1 factors satisfying the spatial isomorphism conjecture”, arXiv:1211.6963 (2012).

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RemarkAI-assistedClaimed by OpenAI. For every epsilon>0, the manuscript claims a universal delta(epsilon)>0 such that unital von Neumann algebras M,N with common identity on any complex Hilbert space and ordinary Kadison–Kastler unit-ball distance less than delta satisfy

uMu^*=N

for a unitary with

\|u-I\|<\epsilon

. The tolerance is uniform over all algebras, representations and Hilbert spaces; an amplified or completely bounded distance assumption is not required.See full solutionHide full solution

Claimed by OpenAI. For every epsilon>0, the manuscript claims a universal delta(epsilon)>0 such that unital von Neumann algebras M,N with common identity on any complex Hilbert space and ordinary Kadison–Kastler unit-ball distance less than delta satisfy $uMu^*=N$ for a unitary with $\|u-I\|<\epsilon$ . The tolerance is uniform over all algebras, representations and Hilbert spaces; an amplified or completely bounded distance assumption is not required.

Scope relative to this problem: The source uses ordinary two-sided Kadison–Kastler distance between operator-norm unit balls of unital von Neumann algebras with a common identity, uniformly over arbitrary complex Hilbert spaces. It claims near-identity spatial conjugacy with a universal epsilon-dependent tolerance, without amplified or completely bounded distance hypotheses.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026/paper.pdf

OpenAI-289-01-Universal-strong-Kadison-Kastler-stability.pdf704,708 bytesOpen

RemarkAI-assistedClaimed by OpenAI. Claims one-sided near inclusions of unital von Neumann algebras on separable complex Hilbert spaces with gaps tending to zero, for which every spatial embedding unitary remains a fixed positive distance from the identity.See full solutionHide full solution

Claimed by OpenAI. Claims one-sided near inclusions of unital von Neumann algebras on separable complex Hilbert spaces with gaps tending to zero, for which every spatial embedding unitary remains a fixed positive distance from the identity.

Scope relative to this problem: Related negative result for a one-sided near-inclusion strengthening: separable-Hilbert gaps tend to zero but every implementing spatial embedding unitary remains a fixed positive distance from the identity. This is not a counterexample to the target two-sided Kadison–Kastler conjugacy assertion.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings-October-5-2026/near-inclusions.pdf

OpenAI-289-02-Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings.pdf313,297 bytesOpen