OP-00968 Open

Kadison's problem on finitely generated *-algebras of operators with finite spectra

Problem. Let $H$ be a Hilbert space and $A \subseteq \mathcal B(H)$ a finitely generated self-adjoint algebra in which every self-adjoint operator has finite spectrum. Must $A$ be finite-dimensional?

Source: R. V. Kadison, Problems on von Neumann algebras (unpublished list, 1967), as recorded by L. Ge (2003); listed as A.1 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).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
Secondary: RA
Keywords: *-algebras, finite spectrum, operator algebras, Peterson's operator algebra problems
Literature checked: Oct. 26, 2026 by clemens-admin

References

@article{ge2003,
  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-a1,
  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 A.1; version of 15 May 2026},
}
@article{mori2023,
  author  = {Mori, Michiya},
  title   = {On regular {$^*$}-algebras of bounded linear operators: a new approach towards a theory of noncommutative {B}oolean algebras},
  journal = {Tohoku Math. J.},
  volume  = {75},
  number  = {3},
  pages   = {423--463},
  year    = {2023},
  doi     = {10.2748/tmj.20220316},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-00968, "Kadison's problem on finitely generated *-algebras of operators with finite spectra", clopen.solutions.