$\mathrm{SL}_{4}(\mathbf{Z})$ is not purely matricial field
Group Theory
2025-03-27 v2 Operator Algebras
Abstract
We prove that every finite dimensional unitary representation of contains a non-zero -invariant vector. As a consequence, there is no sequence of finite-dimensional representations of that gives rise to an embedding of its reduced -algebra into an ultraproduct of matrix algebras.
Cite
@article{arxiv.2312.03220,
title = {$\mathrm{SL}_{4}(\mathbf{Z})$ is not purely matricial field},
author = {Michael Magee and Mikael de la Salle},
journal= {arXiv preprint arXiv:2312.03220},
year = {2025}
}
Comments
v2: 9 pages; details added in proof, and example for SL3 made explicit. To appear in Comptes Rendus Math\'ematique