A note on one-parameter groups of automorphisms (Q749829)

Let \(\{\alpha_ t:\) \(t\in {\mathbb{R}}\}\) and \(\{\beta_ t:\) \(t\in {\mathbb{R}}\}\) be two commuting one-parameter groups of *-automorphisms of a von Neumann algebra M such that \(\alpha_ t+\alpha_{-t}=\beta_ t+\beta_{-t}\) for all \(t\in {\mathbb{R}}\). The authors show that there exists a central projection \(p\in M\) such that \(\alpha_ t=\beta_ t\) on \(M_ p\) and \(\alpha_ t=\beta_{-t}\) on M(1-p) without using the theory of spectral subspaces.