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Math-2822 · mathematics · Analysis

Higher bounded Hochschild cohomology of von Neumann algebras

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every complex von Neumann algebra M and every integer $k\geq 2$, is each bounded k-cocycle with coefficients in M a bounded coboundary, that is $H^k(M,M)=0$, without a complete-boundedness or separability restriction?

Short English statement

For every complex von Neumann algebra M and every integer $k\geq 2$, is each bounded k-cocycle with coefficients in M a bounded coboundary, that is $H^k(M,M)=0$, without a complete-boundedness or separability restriction?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2822-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2822

Manuscript 1 (family 295, 2026-09-23): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Higher bounded Hochschild cohomology of von Neumann algebras

OpenAI's claim

Every bounded Hochschild cocycle of degree two or higher, with self-coefficients on any complex von Neumann algebra, is claimed to admit an actual bounded primitive, settling Kadison--Ringrose cohomology vanishing. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For every complex von Neumann algebra $M$ and integer $k\ge2$, let $C_b^k(M,M)$ be bounded complex multilinear maps. Use the Hochschild differential \[ (df)(a_1,\ldots,a_{k+1})=a_1f(a_2,\ldots,a_{k+1})+\sum_{i=1}^k(-1)^if(a_1,\ldots,a_ia_{i+1},\ldots,a_{k+1})+(-1)^{k+1}f(a_1,\ldots,a_k)a_{k+1}. \] OpenAI claims $df=0$ implies $f=dg$ for some bounded $(k-1)$-cochain, hence $H_b^k(M,M)=0$. The quotient uses the actual coboundary image, without closure; degree one is not the new headline range.

Outline of the argument

Manuscript 1 (family 295, 2026-09-23): A carefully weighted unitary random walk makes averaging in one cocycle variable harmonic. Free-word configurations and a martingale argument prove that harmonic maps must be right-module maps: distinct multipliers on positive and negative words contradict bounded sine-sum estimates. This module identity constructs primitives, after which classical normal-reduction and type-decomposition results extend the argument to all algebras.

Formal verification

For family 295, the scope page documents the following selections. The selected bounded Hochschild primitives cover degrees two and above for arbitrary complex von Neumann algebras, without a type or separability condition. The degree-one derivation result is outside this declaration. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. Comparator files: KadisonRingrose.lean. Declarations: OAI.BoundedHochschild.KadisonRingrose.main_result.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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