Strong uncountable cofinality for unitary groups of von Neumann algebras

TL;DR

This paper proves that the unitary groups of certain von Neumann algebras possess strong uncountable cofinality, providing a new proof for the case of the infinite-dimensional Hilbert space's unitary group.

Contribution

It establishes strong uncountable cofinality for unitary groups of II$_1$ factors and properly infinite von Neumann algebras, offering a shorter proof for the infinite-dimensional case.

Findings

Unitary groups of II$_1$ factors have strong uncountable cofinality.

Unitary groups of properly infinite von Neumann algebras have strong uncountable cofinality.

Provided a shorter proof for the uncountable cofinality of $U(\, ext{ell}^2( ext{N}))$.

Abstract

We show that unitary groups of II$_1$ factors and of properly infinite von Neumann algebras have strong uncountable cofinality. In particular, we obtain a short alternative proof for the strong uncountable cofinality of $U(\ell^2(\mathbb{N}))$, which was first proven by Ricard and Rosendal.