Unitarization of uniformly bounded subgroups in finite von Neumann algebras

Abstract

This note presents 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. In 1974, Vasilescu and Zsido proved this result using the Ryll-Nardzewsky fixed point theorem [Vasilescu and Zsido, "Uniformly bounded groups in finite W*-algebras", Acta Sci. Math. (Szeged) 36 (1974) 189-192]. This new proof involves metric geometric arguments in the non-positively curved space of positive invertible operators of the algebra, which yield a more explicit unitarizer.