Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Studia Mathematica / Online First articles

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.