Universal covering groups of unitary groups of von Neumann algebras

TL;DR

This paper proves that the universal covering group of the unitary group of a II$_1$ von Neumann algebra splits as a direct product, providing a clear algebraic structure and answering a longstanding question about its perfection.

Contribution

It offers a simple proof that the universal cover splits as a direct product, clarifying the algebraic structure of these groups and resolving a question about their perfection.

Findings

Universal cover splits as a direct product of center and unitary group

For II$_1$ factors, the universal cover is isomorphic to $\

,

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.