Browse › Operator algebras › 295 The Kadison–Ringrose cohomology conjecture
Vanishing of higher bounded Hochschild cohomology
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Abstract
We prove that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with values in the algebra itself, has a bounded primitive. Together with the established degree-one inner-derivation theorem, this resolves the Kadison–Ringrose cohomology conjecture positively.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for family 295: The Kadison–Ringrose cohomology conjecture (lean/docs/295.md, release of 6 October 2026). The release's scope note says the formalization covers vanishing of bounded self-coefficient Hochschild cohomology in every degree at least two, with no separability or type restriction, leaves out the degree-one inner-derivation theorem, and links the comparator statement. [family-level]
- OpenAI math release, KadisonRingrose.lean (Comparator challenge statement) (openai/math, commit adc7f12). The challenge file states
main_resultfor aWStarAlgebraM, with cochains taken asContinuousMultilinearMap ℂand the bar-complex differential written out explicitly; as a challenge statement, its proof is left assorry. [family-level] - OpenAI math release, OAI/Analysis/BoundedHochschild/MainResult.lean (solution module). The solution module proves
OAI.BoundedHochschild.KadisonRingrose.main_resultfrombounded_primitivein the imported moduleOAI.Analysis.BoundedHochschild.Main; this file itself contains nosorry. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Erik Christensen, Finite von Neumann algebra factors with property Gamma (arXiv:math/0010139, October 2000). The abstract announces a new proof that the second continuous Hochschild cohomology group of a property Γ II₁ factor with coefficients in itself vanishes, an earlier degree-two case related to the "Prior work" survey in Section 1.1 and not in the paper's bibliography.
Digests and expositions
- Latent Seminar, Bounded Hochschild cohomology: harmonic averaging and rigidity (2026-10-07). This exposition follows the manuscript's harmonic averaging, unitary random walk, ultrapower rigidity and bounded-primitive construction, and explicitly distinguishes its explanation from independent proof verification. [family-level]
- Gradient Damage, Kadison–Ringrose Cohomology Conjecture: Every Twist Is Fake (video and accompanying description, 2026-10-15). The short illustrates the claimed vanishing with a computed primitive for a degree-two cocycle on 3-by-3 matrices and presents the animation as an illustration rather than a proof.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 295
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Vanishing-of-higher-bounded-Hochschild-cohomology-September-23-2026,
author = {{OpenAI}},
title = {{Vanishing of higher bounded Hochschild cohomology}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Vanishing-of-higher-bounded-Hochschild-cohomology-September-23-2026/paper.pdf}{OAI:Vanishing-of-higher-bounded-Hochschild-cohomology-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/295.md. The Lean statements have not been checked by people against the paper's claims.
Family 295: The Kadison–Ringrose cohomology conjecture
Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive. Equivalently, all higher bounded Hochschild cohomology groups vanish, resolving the Kadison–Ringrose conjecture.
This is the only paper in this family.