The Kadison–Kastler conjecture
From OpenAI Aftermath
The Kadison–Kastler conjecture asks whether two operator algebras on the same Hilbert space whose unit balls are close in operator norm must be conjugate by a unitary, and in its strong form whether that unitary can be chosen close to the identity. Three manuscripts in OpenAI's public mathematics collection (github.com/openai/math), grouped there as one family, claim a positive answer to the strong form for von Neumann algebras and two counterexamples at its boundaries: one for one-sided near inclusions and one for the spatial form for norm-separable C*-algebras. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper or a formal proof is not presented here as independent verification.
Background
- As recounted in the introductions of the manuscripts:
- Kadison and Kastler introduced in 1972 a distance between two operator algebras on the same Hilbert space, the Hausdorff distance between their closed unit balls in operator norm, and asked whether closeness in this metric forces spatial isomorphism implemented by a unitary close to the identity.
- Phillips proved spatial isomorphism for type I von Neumann algebras, and Christensen obtained a conjugating unitary close to the identity. Christensen, Johnson, and Raeburn and Taylor established the amenable case, and Christensen developed a one-sided theory of near inclusions for amenable algebras.
- Cameron, Christensen, Sinclair, Smith, White and Wiggins constructed strongly stable nonamenable factors, crossed products of free ergodic actions of SLn(Z) with n ≥ 3 tensored with the hyperfinite II1 factor. For C*-algebras, Christensen, Sinclair, Smith, White and Winter proved spatial isomorphism of sufficiently close algebras on a separable Hilbert space when one of them is norm separable and nuclear, while Johnson's examples rule out, for C*-algebras, a bound on the distance of an implementing unitary from the identity that tends to zero with the algebra distance.
The original results
- Claim (stability). Theorem 1.1 of Universal strong Kadison–Kastler stability states that for every ε > 0 there is δ > 0 such that, for every complex Hilbert space H, any two unital von Neumann algebras M and N on H with the identity of H at Kadison–Kastler distance less than δ satisfy uMu* = N for a unitary u with ‖u − 1‖ < ε. The tolerance is stated to be independent of the algebras, their type, their representations and the Hilbert space; the manuscript states that this resolves the strong Kadison–Kastler conjecture.
- Method, as described: for separably acting finite factors, one argument bounds all matrix amplifications of a commutator by its scalar norm, combining Haagerup's cyclic similarity estimate, Popa's independence theorem for tracial ultraproducts and a free-word randomization; a second argument corrects nearby unitaries to a cocycle over an amenable action and uses double ergodicity to build an isomorphism close to the inclusion. Reductions then give the general case. The manuscript states that its commutator estimate is proved independently of the one in the companion manuscript on Kadison's similarity problem and compares the two in an appendix.
- Claim (one-sided near inclusions). The main theorem of Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings answers negatively the unrestricted one-sided question posed by Cameron, Christensen, Sinclair, Smith, White and Wiggins: there are ε0 > 0 and pairs of von Neumann algebras with a common identity on separable Hilbert spaces whose one-sided near-inclusion gap tends to zero, while every unitary moving the first algebra into the second stays at distance at least ε0 from the identity. The manuscript states that such unitary embeddings exist, that its construction supplies no reverse near inclusion, and that it therefore gives no counterexample to the two-sided conjecture.
- Method, as described: Voiculescu's almost commuting clock and shift matrices, with the Exel–Loring winding-number obstruction, are substituted for a circle parameter in representations of a free group built from the Pytlik–Szwarc representation.
- Claim (separable C*-algebras). Theorem 1.1 of Close Separable C*-Algebras Without Spatial Conjugacy states that for every ε > 0 there are unital, norm-separable C*-algebras with a common identity on a separable complex Hilbert space, at Kadison–Kastler distance less than ε and with the same von Neumann closure, that are not conjugate by any unitary. The manuscript states that this disproves the separable C*-algebraic spatial form of the conjecture, that no nuclearity assumption is imposed, and that it makes no assertion of abstract nonisomorphism and gives no counterexample for von Neumann algebras.
- Method, as described: algebras of norm-convergent sequences whose limits lie in a chosen subalgebra act diagonally on a countable direct sum; their minimal central projections force any conjugating unitary to permute the coordinates, which turns an obstruction for inclusions into an obstruction to spatial conjugacy.
- Sources (openai/math repository, commit
7ef5c0c):- Universal strong Kadison–Kastler stability, dated 23 September 2026;
- Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings, dated 5 October 2026;
- Close Separable C*-Algebras Without Spatial Conjugacy, dated 5 October 2026.
- Formalization, as described by the repository. The scope note lean/docs/289.md lists only the stability manuscript. It states that the formalization proves the strong Kadison–Kastler conjecture uniformly, with a tolerance depending only on ε, and links the comparator statement StrongKadisonKastler.lean. It does not describe a formalization of the two counterexamples.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 28 October 2026) |
|---|---|---|---|---|
| 2026-10-07 | Kadison–Kastler stability and its spatial boundaries, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains the family's three manuscripts as answers to three perturbation questions: universal stability for two-sided perturbations of von Neumann algebras, through matrix-level control and the correction of approximately multiplicative positive maps; the failure of small spatial embeddings for one-sided near inclusions, through an almost-commuting-matrix obstruction; and close norm-closed algebras without spatial conjugacy, through a sequence algebra that records how an inclusion sits inside its ambient algebra. | Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. |
| 2026-10-18 | AI Proved What?, episode 289 Close means a small turn, proved at last?, remade version of 23 October 2026 | Reading (exposition) | Video explainer of the claim that for every ε there is one δ, the same for all pairs of unital von Neumann algebras on any Hilbert space, such that algebras closer than δ are conjugate by a unitary within ε of the identity, together with the two companion counterexamples: a one-sided near inclusion whose embedding unitaries all stay a fixed distance from the identity, and arbitrarily close separable C*-algebras with the same von Neumann closure that are not conjugate by any unitary. The episode states that an explicit tolerance, a bound linear in the distance and a C*-algebra version are not claimed. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as preprints: OpenAI, three papers, 89 pages, dated September 23 and October 5, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: in part: the main theorem, not the counterexamples. The page also states that the site scanned the Lean formalization of the main theorem, about 326,000 lines, without rebuilding it and found no unproved steps, that the two counterexample papers are not formalized and have not been reviewed by OSVerify, and that it found no comment from specialists. |
| 2026-10-20 | Strong Kadison–Kastler stability, a course on Open Mathematics Courses (kokunoyumeto/open-math-courses-public) | Reading (exposition) | Twenty-one lessons: close operator algebras, averaging and the completely positive criterion, boundary means and measurable cocycles, double ergodicity, common tracial position and the stability of finite factors; for infinite factors, rescaled Tomita graphs, infinitesimal normalizers of type III factors, analytic multipliers, a common modular action and common crossed products; centers, measurable assembly and stability on arbitrary Hilbert spaces; and the two counterexamples, through convergent sequences with a relative tensor norm and a free-group deformation, and through a circle family with almost commuting matrices. | The lessons state that they were written by Claude Opus 5.5 (Anthropic) in October 2026 and self-checked by the writing AI; the course lists its status as author self-checked. |
Notes
- The background summarizes the manuscripts' own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
