OP-01048 Solution claimed

Kadison's generator problem for von Neumann algebras

A von Neumann algebra is singly generated if it is generated, as a von Neumann algebra, by one of its elements; equivalently, by two self-adjoint elements. Abelian von Neumann algebras with separable predual are singly generated (von Neumann), and so are those of type I (Pearcy) and those that are properly infinite (Wogen). By reduction theory it suffices to treat II$_1$ factors, and the answer is known to be yes for II$_1$ factors that contain a Cartan subalgebra (Popa), that are not prime (Behncke), that have property $\Gamma$ (Ge and Popa), and for $L(\mathrm{SL}_n(\mathbb Z))$, $n \ge 3$ (Ge and Shen).

Problem. Is every von Neumann algebra with separable predual generated, as a von Neumann algebra, by a single element?

Source: R. V. Kadison, Problems on von Neumann algebras (unpublished list, 1967), as recorded by L. Ge (2003); listed as O.1 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).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: generator problem, Peterson's operator algebra problems, single generation, von Neumann algebras

References

@article{ge2003-o1,
  author  = {Ge, Liming},
  title   = {On ``{P}roblems on von {N}eumann algebras by {R}. {K}adison, 1967''},
  journal = {Acta Math. Sin. (Engl. Ser.)},
  volume  = {19},
  number  = {3},
  pages   = {619--624},
  year    = {2003},
  doi     = {10.1007/s10114-003-0279-x},
}
@misc{peterson-o1,
  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.1; version of 15 May 2026},
}
@article{pearcy1962,
  author  = {Pearcy, Carl},
  title   = {{$W^*$}-algebras with a single generator},
  journal = {Proc. Amer. Math. Soc.},
  volume  = {13},
  number  = {6},
  pages   = {831--832},
  year    = {1962},
  doi     = {10.1090/S0002-9939-1962-0152904-1},
}
@article{wogen1969,
  author  = {Wogen, Warren},
  title   = {On generators for von {N}eumann algebras},
  journal = {Bull. Amer. Math. Soc.},
  volume  = {75},
  number  = {1},
  pages   = {95--99},
  year    = {1969},
  doi     = {10.1090/S0002-9904-1969-12157-9},
}
@article{popa1985,
  author  = {Popa, Sorin},
  title   = {Notes on {C}artan subalgebras in type {II}$_1$ factors},
  journal = {Math. Scand.},
  volume  = {57},
  pages   = {171--188},
  year    = {1985},
}
@article{behncke1972,
  author  = {Behncke, Horst},
  title   = {Generators of finite {$W^*$}-algebras},
  journal = {Tohoku Math. J. (2)},
  volume  = {24},
  pages   = {401--408},
  year    = {1972},
}
@article{gepopa1998,
  author  = {Ge, Liming and Popa, Sorin},
  title   = {On some decomposition properties for factors of type {II}$_1$},
  journal = {Duke Math. J.},
  volume  = {94},
  number  = {1},
  pages   = {79--101},
  year    = {1998},
}
@article{willig1974,
  author  = {Willig, Paul},
  title   = {Generators and direct integral decompositions of {$W^*$}-algebras},
  journal = {Tohoku Math. J.},
  volume  = {26},
  number  = {1},
  pages   = {35--37},
  year    = {1974},
  doi     = {10.2748/tmj/1178241231},
}
@article{geshen2002-o1,
  author  = {Ge, Liming and Shen, Junhao},
  title   = {Generator problem for certain property {T} factors},
  journal = {Proc. Natl. Acad. Sci. USA},
  volume  = {99},
  number  = {2},
  pages   = {565--567},
  year    = {2002},
  doi     = {10.1073/pnas.022593699},
}
@article{dykema2008,
  author  = {Dykema, Ken and Sinclair, Allan and Smith, Roger and White, Stuart},
  title   = {Generators of {II}$_1$ factors},
  journal = {Oper. Matrices},
  volume  = {2},
  number  = {4},
  pages   = {555--582},
  year    = {2008},
  doi     = {10.7153/oam-02-35},
}
@article{popa2021,
  author  = {Popa, Sorin},
  title   = {Tight decomposition of factors and the single generation problem},
  journal = {J. Operator Theory},
  volume  = {85},
  number  = {1},
  pages   = {277--301},
  year    = {2021},
  doi     = {10.7900/jot.2019nov01.2279},
}

Resolution claims

Relative generation and the generator problem for finite factors v1 GitHub ↗

OpenAI

We prove that every type $\mathrm{II}_1$ factor with separable predual is generated by one operator, equivalently by two self-adjoint operators. Combined with the established direct-integral reduction to this case, this gives an affirmative solution of the generator problem for von Neumann algebras with separable predual.

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/main.tex.

Citing this problem

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

OP-01048, "Kadison's generator problem for von Neumann algebras", clopen.solutions.