A II₁ factor with exactly two Cartan subalgebras

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Source: S. Popa, Some open problems in W*-rigidity (2013), who asks for factors with exactly n unitary conjugacy classes of Cartan subalgebras; Problem II in A. Ioana, ICM 2018; listed as C.6 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

DS

Keywords

Cartan subalgebras, II₁ factors, Peterson's operator algebra problems, unitary conjugacy

Literature checked

Oct. 26, 2026 by clemens-admin

References

@misc{popa2013problems-c6,
  author       = {Popa, Sorin},
  title        = {Some open problems in {W}$^*$-rigidity},
  howpublished = {Slides, Paris, June 2013},
  year         = {2013},
  url          = {https://www.math.ucla.edu/~popa/ProblemsJune2013.pdf},
}
@misc{peterson-c6,
  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 C.6; version of 15 May 2026},
}
@article{krogager2017,
  author  = {Krogager, Anna Sofie and Vaes, Stefaan},
  title   = {A class of {II}$_1$ factors with exactly two group measure space decompositions},
  journal = {J. Math. Pures Appl.},
  volume  = {108},
  number  = {1},
  pages   = {88--110},
  year    = {2017},
  doi     = {10.1016/j.matpur.2016.10.011},
}
@article{speelman2012-c6,
  author  = {Speelman, An and Vaes, Stefaan},
  title   = {A class of {II}$_1$ factors with many non conjugate {C}artan subalgebras},
  journal = {Adv. Math.},
  volume  = {231},
  number  = {3-4},
  pages   = {2224--2251},
  year    = {2012},
  doi     = {10.1016/j.aim.2012.07.004},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-01007, "A II₁ factor with exactly two Cartan subalgebras", clopen.solutions.