The Theorem is from Takesaki's book. I was confused by the result which is marked green.
How to derive $sp(\sigma^{\varphi})\subset (2\pi/T)\Bbb Z$.
I tried to search the Propositions and Theorems in the previous Chapter and I cannot find the similar conclusion and I even did not find the precise definition of the $sp(\sigma^{\varphi})$.
As indicated in the book's index, $\operatorname{Sp}(\alpha)$ is the Arveson spectrum of the dynamical system $\{M,G,\alpha\}$, defined on page 313.
The assumption in the second part of the proof is that $M$ is of type III$_\lambda$. By definition (III.1.5), this means that the flow of weights has period $T$ and $T=-\log\lambda$. The explanation for $\log\lambda=\frac{2\pi}T$ seems to stem from the previous section on actions, but I'm not expert.