OP-01060 Solution claimed

The Kadison–Ringrose vanishing cohomology conjecture

Here $H^k(M, M)$ is the bounded (norm-continuous) Hochschild cohomology of $M$ with coefficients in $M$. For $k = 1$ it vanishes by the theorem of Kadison and Sakai that every derivation of a von Neumann algebra is inner.

Problem. Is $H^k(M, M) = 0$ for every von Neumann algebra $M$ and every $k \ge 1$?

Source: R. V. Kadison and J. R. Ringrose (1967; Cohomology of operator algebras I, 1971); listed as O.6 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).

Imported from math.uwaterloo.ca/~j37peter/problems.html.

Classification

Primary: OA — Operator Algebras
Keywords: derivations, Hochschild cohomology, Peterson's operator algebra problems, von Neumann algebras

References

@article{kadison1971,
  author  = {Kadison, Richard V. and Ringrose, John R.},
  title   = {Cohomology of operator algebras {I}. {T}ype {I} von {N}eumann algebras},
  journal = {Acta Math.},
  volume  = {126},
  pages   = {227--243},
  year    = {1971},
  doi     = {10.1007/BF02392032},
}
@misc{peterson-o6,
  author       = {Peterson, Jesse},
  title        = {Open problems in operator algebras},
  howpublished = {Web page},
  year         = {2026},
  url          = {https://www.math.uwaterloo.ca/~j37peter/problems.html},
  note         = {Problem O.6; version of 15 May 2026},
}
@article{johnson1972,
  author  = {Johnson, B. E. and Kadison, Richard V. and Ringrose, John R.},
  title   = {Cohomology of operator algebras. {III}. {R}eduction to normal cohomology},
  journal = {Bull. Soc. Math. France},
  volume  = {100},
  pages   = {73--96},
  year    = {1972},
  doi     = {10.24033/bsmf.1731},
}
@article{christensen2003,
  author  = {Christensen, Erik and Pop, Florin and Sinclair, Allan and Smith, Roger},
  title   = {Hochschild cohomology of factors with property {$\Gamma$}},
  journal = {Ann. of Math.},
  volume  = {158},
  number  = {2},
  pages   = {635--659},
  year    = {2003},
  doi     = {10.4007/annals.2003.158.635},
}
@article{sinclair1998,
  author  = {Sinclair, Allan M. and Smith, Roger R.},
  title   = {Hochschild cohomology for von {N}eumann algebras with {C}artan subalgebras},
  journal = {Amer. J. Math.},
  volume  = {120},
  number  = {5},
  pages   = {1043--1057},
  year    = {1998},
  doi     = {10.1353/ajm.1998.0042},
}
@article{qian2015,
  author  = {Qian, Wenhua and Shen, Junhao},
  title   = {Hochschild cohomology of type {II}$_1$ von {N}eumann algebras with property {$\Gamma$}},
  journal = {Oper. Matrices},
  volume  = {9},
  number  = {3},
  pages   = {507--543},
  year    = {2015},
  doi     = {10.7153/oam-09-31},
}

Resolution claims

Vanishing of higher bounded Hochschild cohomology v1 GitHub ↗

OpenAI

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.

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-29	Needs verification → Solution claimed	—	Recorded here: math @ 7ef5c0c, build/paper.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-01060, "The Kadison–Ringrose vanishing cohomology conjecture", clopen.solutions.