one parameter-family of unitaries lie in the commutant of a von Neuamann algebra

Question

The screenshot is from Herman's paper. I have two questions in the proof of Prolosition 3.

Question 1: If $M_{\varphi}\subset M_{\psi}$, then $v_t\in M_{\psi}'\cap M$. For any $x\in M_{\psi}$, we want to prove $xv_t=v_tx$. Since $x\in M_{\psi}$, we have $x\psi=\psi x$, then $\psi(v_t x)=\psi(x v_t)$. How to conclude $xv_t=v_tx$?

Question 2. How to prove $\sigma_t^{\psi}=v_t\sigma_t^{\psi}(y)v_t^{*}$? Is $\sigma_t^{\psi}(y)\in M_{\psi}$?

Answer 1

1. Let $x\in M_\varphi$. Then $$ x=\sigma_t^\psi(x)=v_t^*\sigma_t^\varphi(x)v_t=v_t^*xv_t. $$ So $x\in M_\varphi'\cap M=M_\psi'\cap M$.

2. If $y\in M_\psi$, since $v_t\in M_\psi'\cap M$, $$ \sigma_t^\varphi(y)=v_t\sigma_t^\psi(y)v_t^*=v_tyv_t^*=y. $$