Unitarization of uniformly bounded subgroups in finite von Neumann algebras

TL;DR

This paper provides a new proof that uniformly bounded groups of invertible elements in finite von Neumann algebras are similar to unitary groups, using metric geometric methods in non-positively curved spaces.

Contribution

It introduces a novel proof technique based on metric geometry, differing from the classical fixed point theorem approach.

Findings

Every uniformly bounded group in a finite von Neumann algebra is similar to a unitary group.

The proof employs metric geometric arguments in the space of positive invertible operators.

Provides an alternative perspective to classical fixed point methods.

Abstract

This note will present a new proof of the fact that every uniformly bounded group of invertible elements in a finite von Neumann algebra is similar to a unitary group. The proof involves metric geometric arguments in the non-positively curved space of positive invertible operators of the algebra; in 1974 Vasilescu and Zsido proved this result using the Ryll-Nardzewsky fixed point theorem.