Universal covering groups of unitary groups of von Neumann algebras
Volume 282 / 2025
Studia Mathematica 282 (2025), 89-100
MSC: Primary 46L10; Secondary 46L05, 46L80
DOI: 10.4064/sm240829-14-1
Published online: 28 February 2025
Abstract
We give a short and simple proof, which utilizes the pre-determinant of P. de la Harpe and G. Skandalis, that the universal covering group of the unitary group of a $\mathrm{II}_1$ von Neumann algebra $\mathcal M $, when equipped with the norm topology, splits algebraically as the direct product of the self-adjoint part of its centre and the unitary group $U(\mathcal M )$. Thus, when $\mathcal M $ is a $\mathrm{II} _1$ factor, the universal covering group of $U(\mathcal M )$ is algebraically isomorphic to the direct product $\mathbb R \times U(\mathcal M )$. In particular, the question of P. de la Harpe and D. McDuff of whether the universal cover of $U(\mathcal M )$ is a perfect group is answered in the negative.
