A note on one-parameter groups of automorphisms

Abstract

Let {αt:t∈R} and {βt:t∈R} be two commuting one-parameter groups of ∗-automorphisms of a von Neumann algebra M such that αt+α−t=βt+β−t for all t∈R. The purpose of this note is to provide a simple and short proof of the central decomposition result: αt=βt on Mp and a αt=β−t on M(1−p) for a central projection p∈M, without using the theory of spectral subspaces.