OP-00978 Open

Unique Cartan subalgebra for SLₙ(ℤ) acting on the n-torus

Here $\mathrm{SL}_n(\mathbb Z)$ acts on the torus $\mathbb T^n = \mathbb R^n / \mathbb Z^n$ by linear automorphisms, preserving Haar measure.

Problem. Does $L^\infty(\mathbb T^n) \rtimes \mathrm{SL}_n(\mathbb Z)$ have a unique Cartan subalgebra up to unitary conjugacy for every $n \ge 3$?

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

References

@misc{popa2013problems-c3,
  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-c3,
  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.3; version of 15 May 2026},
}
@article{boutonnet2021,
  author  = {Boutonnet, R{\'e}mi and Ioana, Adrian and Peterson, Jesse},
  title   = {Properly proximal groups and their von {N}eumann algebras},
  journal = {Ann. Sci. {\'E}c. Norm. Sup{\'e}r.},
  volume  = {54},
  number  = {2},
  pages   = {445--482},
  year    = {2021},
  doi     = {10.24033/asens.2462},
}
@article{ioana2012,
  author  = {Ioana, Adrian},
  title   = {Uniqueness of the group measure space decomposition for {P}opa's $\mathcal{HT}$ factors},
  journal = {Geom. Funct. Anal.},
  volume  = {22},
  number  = {3},
  pages   = {699--732},
  year    = {2012},
  doi     = {10.1007/s00039-012-0178-3},
}
@article{furman1999,
  author  = {Furman, Alex},
  title   = {Orbit equivalence rigidity},
  journal = {Ann. of Math.},
  volume  = {150},
  number  = {3},
  pages   = {1083--1108},
  year    = {1999},
  doi     = {10.2307/121063},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-00978, "Unique Cartan subalgebra for SLₙ(ℤ) acting on the n-torus", clopen.solutions.