Unitarization of uniformly bounded subgroups in finite von Neumann algebras

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.