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The strong Kadison-Kastler conjecture

For von Neumann algebras M,N⊆B(H)M,N\subseteq B(H) the Kadison-Kastler distance d(M,N)d(M,N) is the Hausdorff distance between their unit balls in operator norm. Kadison and Kastler (1972) proposed that sufficiently close von Neumann algebras should be spatially isomorphic via a unitary close to the identity. This was proved for injective algebras (Christensen, Johnson, Raeburn-Taylor) and for certain crossed-product II1\mathrm{II}_1 factors (Cameron et al.). Is there, for every ε>0\varepsilon>0, a δ>0\delta>0 such that any two von Neumann algebras on the same Hilbert space with d(M,N)<δd(M,N)<\delta satisfy uMu∗=NuMu^*=N for a unitary uu with ∥u−1∥<ε\|u-1\|<\varepsilon?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Operator algebras; perturbation theory of von Neumann algebras
Posed by
R. V. Kadison and D. Kastler, Perturbations of von Neumann algebras. I. Stability of type, Amer. J. Math. 94 (1972)
Year posed
1972
Years open
54y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
45 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every ε>0\varepsilon>0 there is δ(ε)>0\delta(\varepsilon)>0, independent of the algebras, their type, representations and the Hilbert space, such that unital von Neumann algebras M,N⊆B(H)M,N\subseteq B(H) with d(M,N)<δd(M,N)<\delta satisfy uMu∗=NuMu^*=N with ∥u−1∥<ε\|u-1\|<\varepsilon. The proof combines amplification estimates, a modular and core analysis for type III, and reductions across types. It concerns von Neumann algebras only; the companions show that one-sided near inclusions and close separable C*-algebras behave differently. No explicit δ(ε)\delta(\varepsilon) is given.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the paper states it with the 1972 reference; it claims a tolerance δ(ε)\delta(\varepsilon) uniform over all unital von Neumann algebras, representations and Hilbert spaces. The proof was not refereed. The challenge ComparatorChallenges/StrongKadisonKastler.lean is not in lean/formalization.yaml; it is found through lean/docs/289.md and its solution module OAI.Analysis.KadisonKastler.StrongStability exists at the pinned commit. Its statement was read here: for every ε>0\varepsilon>0 there is δ>0\delta>0 such that for every complex Hilbert space HH and von Neumann algebras M,NM,N on HH with Hausdorff distance of unit balls below δ\delta there is a unitary vv with vMv∗=NvMv^*=N and ∥v−1∥<ε\|v-1\|<\varepsilon. This states the headline. Not rebuilt here. The paper cites the release's similarity companion for a related commutator estimate.

Sources

Changelog1 change

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