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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.

Unitarization of uniformly bounded subgroups in finite von Neumann algebras