OP-01062 Needs verification

Kadison's problem on orthonormal bases of unitaries in II₁ factors

Needs verification: A solution has been claimed in the literature; held until the record of the claim is listed here.

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For a II$_1$ factor $M$ with trace $\tau$, $L^2(M, \tau)$ is the completion of $M$ in the inner product $\langle x, y \rangle = \tau(y^* x)$.

Problem. Does every II$_1$ factor $M$ have an orthonormal basis of $L^2(M, \tau)$ consisting of unitaries of $M$?

Source: R. V. Kadison, Problems on von Neumann algebras (unpublished list, 1967), as recorded by L. Ge (2003); listed as S.2 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).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: II₁ factors, orthonormal bases, Peterson's operator algebra problems, unitaries

References

@article{ge2003-s2,
  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-s2,
  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.2; version of 15 May 2026},
}
@article{akemann2014,
  author  = {Akemann, Charles and Weaver, Nik},
  title   = {A {L}yapunov-type theorem from {K}adison--{S}inger},
  journal = {Bull. Lond. Math. Soc.},
  volume  = {46},
  number  = {3},
  pages   = {517--524},
  year    = {2014},
  doi     = {10.1112/blms/bdu005},
}

Citing this problem

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OP-01062, "Kadison's problem on orthonormal bases of unitaries in II₁ factors", clopen.solutions.