OP-01049 Solution claimed

Is every von Neumann algebra with property (T) singly generated?

Factors with property (T) are separable, so this is a case of Kadison's generator problem. Hutchcroft and Pete proved that every countably infinite group with property (T) has cost $1$.

Problem. Is every von Neumann algebra with property (T) generated, as a von Neumann algebra, by a single element?

Source: Listed without attribution as S.1 in J. Peterson, Open problems in operator algebras (version of 15 May 2026), as a case of Kadison's generator problem.

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

Classification

Primary: OA — Operator Algebras
Keywords: generator problem, Peterson's operator algebra problems, property (T), single generation

References

@misc{peterson-s1,
  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 S.1; version of 15 May 2026},
}
@article{hutchcroft2020-s1,
  author  = {Hutchcroft, Tom and Pete, G{\'a}bor},
  title   = {Kazhdan groups have cost 1},
  journal = {Invent. Math.},
  volume  = {221},
  number  = {3},
  pages   = {873--891},
  year    = {2020},
  doi     = {10.1007/s00222-020-00967-6},
}
@article{geshen2002-s1,
  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},
}

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 It will not change if the problem is solved, reclassified or withdrawn.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-01049, "Is every von Neumann algebra with property (T) singly generated?", clopen.solutions.