The above statement is from Takesaki's book 'Theory of Operator Algebras vol II'(See page 375).
I was struggled with verifying the conclusion $N=M_{\varphi}$.
How to check that for any $x\in N$, we have $\sigma_t^{\varphi}(x)=x,\forall t\in \Bbb R$ and for $y\in M_{\varphi}$, we have $y\in N$?