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The Kadison-Ringrose cohomology conjecture

For a complex von Neumann algebra MM, let Cbk(M,M)C_b^k(M,M) be the bounded complex kk-linear maps Mk→MM^k\to M (operator-norm boundedness, no complete boundedness), with the Hochschild differential dd, and let Hbk(M,M)=ker⁡d/im⁡dH_b^k(M,M)=\ker d/\operatorname{im} d, the image taken without closure. In degree one vanishing is the Kadison-Sakai theorem that every derivation of MM is inner. Kadison and Ringrose proved vanishing for type I and hyperfinite algebras, Christensen, Effros and Sinclair whenever the type II1\mathrm{II}_1 summand absorbs the hyperfinite II1\mathrm{II}_1 factor, and later work handled Cartan subalgebras and property Γ\Gamma. Is Hbk(M,M)=0H_b^k(M,M)=0 for every complex von Neumann algebra MM and every k≥1k\ge1?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Operator algebras: bounded Hochschild cohomology
Posed by
Richard V. Kadison and John R. Ringrose (Cohomology of operator algebras I, II, 1971); Sinclair and Smith trace the self-coefficient formulation to 1967
Year posed
1967
Years open
59y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
42 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims Theorem 1.1: for every complex von Neumann algebra MM and every k≥2k\ge2, every bounded kk-cocycle f∈Cbk(M,M)f\in C_b^k(M,M) with df=0df=0 equals dgdg for some bounded g∈Cbk−1(M,M)g\in C_b^{k-1}(M,M), so Hbk(M,M)=0H_b^k(M,M)=0. Combined with the Kadison-Sakai theorem in degree one this is the full conjecture. The key new input is a Liouville theorem for harmonic maps under a single unitary random walk built from Popa's free-independent sequences in type II1\mathrm{II}_1 algebras. It concerns ordinary bounded cohomology with coefficients in MM itself; it says nothing new about coefficients in B(H)B(H) or about completely bounded cohomology beyond what follows.

What the AI did

Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against the conjecture as the manuscript states it (vanishing of Hbk(M,M)H_b^k(M,M) for all complex von Neumann algebras and all k≥1k\ge1). The proof was not refereed. Theorem 1.1 covers every degree k≥2k\ge2 with no separability or type restriction; degree one is the classical Kadison-Sakai inner-derivation theorem, which the paper cites rather than reproves. Lean-checked on the release's own Comparator challenge KadisonRingrose together with its solution module OAI.Analysis.BoundedHochschild.MainResult, both fetched at the pinned commit; the challenge is not listed in the release's formalization catalogue (lean/formalization.yaml), the statement was read here but not independently audited, and the development was not rebuilt here. The formal statement (OAI.BoundedHochschild.KadisonRingrose.main_result) takes any C*-algebra with a W*-algebra (predual) structure and says every bounded multilinear cocycle of degree n+2 with values in the algebra is the differential of a bounded cochain of degree n+1, which is the headline claim for all degrees at least two.

Sources

Changelog1 change

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