OP-01000 Open

Is L(SL₃(ℤ)) pseudomatricial?

A II$_1$ factor is pseudomatricial if some ultrapower of it is isomorphic to an ultraproduct of matrix algebras.

Problem. Is $L(\mathrm{SL}_3(\mathbb Z))$ pseudomatricial? Is any II$_1$ factor with property (T) pseudomatricial?

Source: Nearest written source: S. Kunnawalkam Elayavalli, 50 open problems: ultraproduct II₁ factors (2025), Problem 4, which asks whether some group von Neumann algebra is pseudocompact (pseudomatricial); listed, dated 2025, as U.11 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: LO
Keywords: matrix algebras, Peterson's operator algebra problems, property (T), pseudomatricial factors, ultraproducts
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{kunnawalkam2025-u11,
  author        = {Kunnawalkam Elayavalli, Srivatsav},
  title         = {50 open problems: ultraproduct {II}$_1$ factors},
  year          = {2025},
  eprint        = {2511.20377},
  archivePrefix = {arXiv},
  note          = {Problem 4; Problem 3 is the case of L(F_n); v3, 25 Aug 2026},
}
@misc{peterson-u11,
  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 U.11; version of 15 May 2026},
}
@misc{peterson2026-u11,
  author        = {Peterson, Jesse},
  title         = {On ultraproduct approximations and property {(T)} factors},
  year          = {2026},
  eprint        = {2605.16669},
  archivePrefix = {arXiv},
  note          = {v2, 1 Oct 2026; Theorem 1.2: L(F_n) is not pseudomatricial},
}
@article{farah2014-u11,
  author  = {Farah, Ilijas and Hart, Bradd and Sherman, David},
  title   = {Model theory of operator algebras {III}: elementary equivalence and {II}$_1$ factors},
  journal = {Bull. Lond. Math. Soc.},
  volume  = {46},
  number  = {3},
  pages   = {609--628},
  year    = {2014},
  doi     = {10.1112/blms/bdu012},
}

Citing this problem

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OP-01000, "Is L(SL₃(ℤ)) pseudomatricial?", clopen.solutions.