∑math20

Operator algebras · Result 295

The Kadison–Ringrose cohomology conjecture

1 paperLean

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.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 1

First page of Vanishing of higher bounded Hochschild cohomology

Vanishing of higher bounded Hochschild cohomology

OpenAI · September 23, 2026

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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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}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves vanishing of bounded Hochschild cohomology in every degree at least two for a complex von Neumann algebra with coefficients in itself. Every bounded multilinear cocycle of such a degree is the Hochschild differential of a bounded multilinear cochain one degree lower. No separability or type restriction is imposed. The degree-one inner-derivation theorem is outside this selected statement.

Comparator links

ResultComparator statement
Bounded primitives for all Hochschild cocycles of degree at least twoKadisonRingrose.lean

Scope note on GitHub →