OP-00963 Open

Are the radial and generator masas of L(F₂) conjugate by an automorphism?

Let $\mathbb F_2$ be the free group on $a$ and $b$, and let $u_g \in L(\mathbb F_2)$ be its canonical unitaries. The generator masa $\{u_a\}''$ and the radial masa $\{u_a + u_a^* + u_b + u_b^*\}''$ are maximal abelian von Neumann subalgebras of $L(\mathbb F_2)$.

Problem. Is there an automorphism $\theta$ of $L(\mathbb F_2)$ with $\theta(\{u_a + u_a^* + u_b + u_b^*\}'') = \{u_a\}''$?

Source: Cameron, Fang, Ravichandran and White (2010), Question 1.1; listed as S.3 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: free group factors, maximal abelian subalgebras, Peterson's operator algebra problems, radial masa
Literature checked: Oct. 26, 2026 by clemens-admin

References

@article{cameron2010,
  author  = {Cameron, Jan and Fang, Junsheng and Ravichandran, Mohan and White, Stuart},
  title   = {The radial masa in a free group factor is maximal injective},
  journal = {J. Lond. Math. Soc.},
  volume  = {82},
  number  = {3},
  pages   = {787--809},
  year    = {2010},
  doi     = {10.1112/jlms/jdq052},
}
@misc{peterson-s3,
  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.3; version of 15 May 2026},
}
@article{radulescu1991,
  author  = {Radulescu, Florin},
  title   = {Singularity of the radial subalgebra of {$\mathcal L(F_N)$} and the {P}uk{\'a}nszky invariant},
  journal = {Pacific J. Math.},
  volume  = {151},
  number  = {2},
  pages   = {297--306},
  year    = {1991},
  doi     = {10.2140/pjm.1991.151.297},
}
@article{boutonnet2026,
  author  = {Boutonnet, R{\'e}mi and Popa, Sorin},
  title   = {Maximal amenable {MASA}s of radial type in the free group factors},
  journal = {Proc. Amer. Math. Soc.},
  year    = {2026},
  doi     = {10.1090/proc/17713},
}
@misc{boschert2024,
  author        = {Boschert, Nicholas and Davis, Ethan and Hiatt, Patrick},
  title         = {A class of freely complemented von {N}eumann subalgebras of {$L\mathbb F_n$}},
  year          = {2024},
  eprint        = {2411.05136},
  archivePrefix = {arXiv},
}
@article{dykema2013,
  author  = {Dykema, Ken and Mukherjee, Kunal},
  title   = {Measure-multiplicity of the {L}aplacian masa},
  journal = {Glasg. Math. J.},
  volume  = {55},
  number  = {2},
  pages   = {285--292},
  year    = {2013},
  doi     = {10.1017/S001708951200050X},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-00963, "Are the radial and generator masas of L(F₂) conjugate by an automorphism?", clopen.solutions.