Universal covering groups of unitary groups of von Neumann algebras

Abstract

We give a short and simple proof, utilizing the pre-determinant of P. de la Harpe and G. Skandalis, that the universal covering group of the unitary group of a 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 center and the unitary group $U(\mathcal{M})$. Thus, when $\mathcal{M}$ is a 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.