Result 295, Operator algebras

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.

Lean formalization Proof

The bigger picture

Why it matters

Cohomology tests whether algebraic compatibility conditions hide an obstruction to solving an equation. This manuscript reports that, for a broad class of operator algebras, every bounded higher-degree obstruction of a specific kind disappears.

What changes?

The claim covers every complex von Neumann algebra, an algebra of bounded operators closed under adjoints and suitable limits. A bounded Hochschild cocycle is a bounded multilinear map satisfying a compatibility equation determined by multiplication. For every degree at least two, with the map taking values in the algebra itself, the manuscript reports a bounded primitive: a map of one lower degree whose Hochschild boundary is the original cocycle. Equivalently, all higher bounded Hochschild cohomology groups with these coefficients vanish.

What does that help mathematicians do?

A researcher encountering such a cocycle could therefore deduce that its equation has a bounded primitive, rather than needing a separate existence argument for that algebra. Combined with the established degree-one inner-derivation theorem, the reported result makes these cohomology groups zero in every positive degree. Consequently, these particular groups cannot distinguish complex von Neumann algebras from one another, sharply limiting their use as algebraic invariants.

Are there practical applications?

The immediate value is foundational: the claim settles whether these compatibility equations admit bounded solutions throughout this class of operator algebras. It would let subsequent arguments rely on existence rather than assume it. The supplied abstract does not describe a computational procedure or quantitative estimates for constructing the primitives.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Vanishing of higher bounded Hochschild cohomology

September 23, 2026 28 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/295.md.

The Kadison–Ringrose cohomology conjecture

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

Result Comparator statement
Bounded primitives for all Hochschild cocycles of degree at least two KadisonRingrose.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.