Universal covering groups of unitary groups of von Neumann algebras

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 II₁ von Neumann algebra 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(M). Thus, when M is a II₁ factor, the universal covering group of U(M) is algebraically isomorphic to the direct product R × U(M). In particular, the question of P. de la Harpe and D. McDuff of whether the universal cover of U(M) is a perfect group is answered in the negative.