Unitary representations of higher-rank lattices in the hyperfinite II₁ factor
Let $R$ be the hyperfinite II$_1$ factor and $\mathcal U(R)$ its unitary group. When $\Gamma$ has property (T), every homomorphism $\Gamma \to \mathcal U(R)$ has precompact image in the strong operator topology (Robertson; another proof by Valette).
Problem. Let $\Gamma$ be an irreducible lattice in a higher-rank semisimple real Lie group, and $\pi : \Gamma \to \mathcal U(R)$ a homomorphism. Must $\pi(\Gamma)$ be precompact in the strong operator topology?
Source: Listed, as an extension to all higher-rank lattices of Robertson's theorem for property (T) groups, as H.4 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
- GR, RT
- Keywords
- hyperfinite II₁ factor, lattices in Lie groups, Peterson's operator algebra problems, rigidity, unitary representations
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@article{robertson1993,
author = {Robertson, A. Guyan},
title = {Property {(T)} for {II}$_1$ factors and unitary representations of {K}azhdan groups},
journal = {Math. Ann.},
volume = {296},
number = {1},
pages = {547--555},
year = {1993},
doi = {10.1007/BF01445119},
}
@article{valette1997,
author = {Valette, Alain},
title = {Amenable representations and finite injective von {N}eumann algebras},
journal = {Proc. Amer. Math. Soc.},
volume = {125},
number = {6},
pages = {1841--1843},
year = {1997},
doi = {10.1090/S0002-9939-97-03754-4},
}
@misc{peterson-h4,
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 H.4; version of 15 May 2026},
}
@article{creutz2024,
author = {Creutz, Darren and Peterson, Jesse},
title = {Character rigidity for lattices and commensurators},
journal = {Amer. J. Math.},
volume = {146},
number = {3},
pages = {687--711},
year = {2024},
doi = {10.1353/ajm.2024.a928322},
}
@article{bader2022,
author = {Bader, Uri and Boutonnet, R{\'e}mi and Houdayer, Cyril and Peterson, Jesse},
title = {Charmenability of arithmetic groups of product type},
journal = {Invent. Math.},
volume = {229},
number = {3},
pages = {929--985},
year = {2022},
doi = {10.1007/s00222-022-01117-w},
}
@misc{dogon2025,
author = {Dogon, Alon and Glasner, Michael and Gorfine, Yuval and Hanany, Liam and Levit, Arie},
title = {Non-uniform higher-rank lattices are character rigid},
year = {2025},
eprint = {2507.21862},
archivePrefix = {arXiv},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00976, "Unitary representations of higher-rank lattices in the hyperfinite II₁ factor", clopen.solutions.