View source for The Kadison–Ringrose cohomology conjecture

'''The Kadison–Ringrose cohomology conjecture''' asks whether the bounded Hochschild cohomology of every von Neumann algebra with coefficients in itself vanishes in all positive degrees: every bounded multilinear cocycle should be the coboundary of a bounded cochain. A manuscript in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 23 September 2026, claims vanishing in every degree at least two, which together with the classical inner-derivation theorem in degree one would settle the conjecture positively. This page summarizes the published claim 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 introduction of the manuscript:
** In degree one the question is whether every derivation of a von Neumann algebra into itself is inner, which is the theorem of Kadison and Sakai. The higher-degree theory was developed by Johnson, Kadison and Ringrose; Sinclair and Smith trace the self-coefficient formulation of the conjecture to 1967.
** Kadison and Ringrose proved vanishing for type I and hyperfinite von Neumann algebras, and Johnson, Kadison and Ringrose identified continuous cohomology with separately normal cohomology. The completely bounded methods of Christensen, Effros and Sinclair gave vanishing whenever the type II<sub>1</sub> central summand absorbs the hyperfinite II<sub>1</sub> factor, including all algebras without a type II<sub>1</sub> summand.
** For the remaining finite algebras, vanishing was known for algebras with a Cartan subalgebra (Sinclair and Smith for separable predual; Cameron for type II<sub>1</sub> factors without separability), for type II<sub>1</sub> factors with property Γ (Christensen, Pop, Sinclair and Smith), for a generalized property Γ with separable predual (Qian and Shen), and in degree two for tensor products of two type II<sub>1</sub> algebras (Pop and Smith).

== The original result ==
* '''Claim.''' Theorem 1.1 of ''Vanishing of higher bounded Hochschild cohomology'' states that for every complex von Neumann algebra ''M'' and every ''k'' ≥ 2, every bounded ''k''-linear map ''f'' from ''M''<sup>''k''</sup> to ''M'' with ''df'' = 0 is of the form ''f'' = ''dg'' for a bounded (''k'' − 1)-linear map ''g''. Boundedness is ordinary operator-norm boundedness, without a complete-boundedness condition, and the image is taken without closure. The manuscript states that, together with the inner-derivation theorem, this resolves the Kadison–Ringrose cohomology conjecture positively.
* ''Method, as described:'' in a type II<sub>1</sub> algebra with separable predual, averaging one cocycle input along a single symmetric unitary random walk produces a harmonic map. A Liouville theorem shows that such a map commutes with right multiplication; its proof uses free-independent unitary configurations from Popa's work, a harmonic martingale and a rigidity argument in a tracial ultrapower. The resulting right module map yields a primitive with norm at most that of the cocycle. Compactness then removes the separability assumption, and a central decomposition together with the classical vanishing theorem gives the general case.
* '''Source''' (openai/math repository, commit <code>7ef5c0c</code>): [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/Vanishing-of-higher-bounded-Hochschild-cohomology-September-23-2026 Vanishing of higher bounded Hochschild cohomology], dated 23 September 2026.
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/lean/docs/295.md lean/docs/295.md] states that the formalization proves vanishing of bounded Hochschild cohomology in every degree at least two for a complex von Neumann algebra with coefficients in itself, with no separability or type restriction, and that the degree-one inner-derivation theorem is outside the selected statement. It links the comparator statement KadisonRingrose.lean.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 28 October 2026)
|-
| 2026-10-07 || [https://lowrank.github.io/latent-seminar/OpenAI-Math/295-bounded-hochschild-cohomology-harmonic-rigidity ''Bounded Hochschild cohomology: harmonic averaging and rigidity''], an article of the Latent Seminar blog at lowrank.github.io || Reading (exposition) || Explains the manuscript's proof that bounded Hochschild cocycles of degree at least two on a von Neumann algebra have bounded primitives: harmonic averaging of one cocycle input along a unitary random walk, a Liouville theorem that makes the harmonic map a right module map, rigidity in a tracial ultrapower, and the construction of the primitive. || 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 295 [https://aiprovedwhat.com/episodes/295-kadison-ringrose/ Every equation solvable?], remade version of 23 October 2026 || Reading (exposition) || Video explainer of the claim that for every complex von Neumann algebra and every degree at least two, every bounded cocycle with coefficients in the algebra is the coboundary of a bounded map with values in the algebra, so that together with the Kadison–Sakai theorem in degree one all positive degrees vanish. The episode states that other coefficient modules, C*-algebras and one constant for all algebras 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 a preprint: OpenAI, 28 pages, dated September 23, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: the main theorem, with the classical inputs. The page also states that the formalization matches the paper in about 63,000 lines that also cover the classical inputs, that the site scanned it without building it and found no unproved steps, that OSVerify has not reviewed the paper, and that it found no expert comment or reported error.
|-
| 2026-10-19 || [https://kokunoyumeto.github.io/open-math-courses-public/courses/bounded-hochschild-cohomology-of-von-neumann-algebras/ ''Bounded Hochschild cohomology of von Neumann algebras''], a course on Open Mathematics Courses ([https://github.com/kokunoyumeto/open-math-courses-public kokunoyumeto/open-math-courses-public]) || Reading (exposition) || Five lessons: the cohomology problem; rigidity of first-letter identities on free Haar families; a random walk on the unitary group whose long products are asymptotically free; the Liouville theorem for its harmonic maps; and bounded primitives by averaging, with the removal of separability and an explicit central homotopy. || 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 manuscript's 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.

[[Category:OpenAI results]]