OP-00977 Open

Unique Cartan subalgebra for Bernoulli crossed products

Problem. Let $\Gamma$ be a nonamenable group and $(X_0, \mu_0)$ a nontrivial standard probability space. Does the crossed product $L^\infty\big(\prod_{g \in \Gamma} (X_0, \mu_0)\big) \rtimes \Gamma$ of the Bernoulli shift have a unique Cartan subalgebra up to unitary conjugacy?

Source: Conjecture of S. Popa, Some open problems in W*-rigidity (2013); listed as C.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: DS, GR
Keywords: Bernoulli actions, Cartan subalgebras, group measure space construction, Peterson's operator algebra problems
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{popa2013problems-c1,
  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-c1,
  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.1; version of 15 May 2026},
}
@article{ding2025,
  author  = {Ding, Changying},
  title   = {A unique {C}artan subalgebra result for {B}ernoulli actions of weakly amenable groups},
  journal = {J. Funct. Anal.},
  volume  = {288},
  number  = {10},
  pages   = {110852},
  year    = {2025},
  doi     = {10.1016/j.jfa.2025.110852},
}
@article{popavaes2014acta-c1,
  author  = {Popa, Sorin and Vaes, Stefaan},
  title   = {Unique {C}artan decomposition for {II}$_1$ factors arising from arbitrary actions of free groups},
  journal = {Acta Math.},
  volume  = {212},
  number  = {1},
  pages   = {141--198},
  year    = {2014},
  doi     = {10.1007/s11511-014-0110-9},
}
@article{ioana2011,
  author  = {Ioana, Adrian},
  title   = {{W}$^*$-superrigidity for {B}ernoulli actions of property {(T)} groups},
  journal = {J. Amer. Math. Soc.},
  volume  = {24},
  number  = {4},
  pages   = {1175--1226},
  year    = {2011},
  doi     = {10.1090/S0894-0347-2011-00706-6},
}
@inproceedings{ioana2018,
  author    = {Ioana, Adrian},
  title     = {Rigidity for von {N}eumann algebras},
  booktitle = {Proceedings of the International Congress of Mathematicians (Rio de Janeiro 2018), Vol. III},
  publisher = {World Sci. Publ.},
  pages     = {1639--1672},
  year      = {2018},
  doi       = {10.1142/9789813272880_0111},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-00977, "Unique Cartan subalgebra for Bernoulli crossed products", clopen.solutions.