A theorem in Takesaki's book (Vol 2, Chapter VIII)

Question

The above screenshot is from Takesaki's book.

I have two questions.

1. In the proof of Theorem 3.2. The author mention that if $\psi$ is not faithful, then we can choose $\psi^{'}$ om $M$ such that $s(\psi^{'}= 1-s(\psi)$. How to prove the existence of the weight $\psi^{'}$?

2. The proof of the last statement of the theorem is not given. I wonder how to construct $\xi\in H_{\varphi} $ such that the $\omega(x)=(\pi_{\varphi}(x)\xi,\xi)$?

Answer 1

1. You define $\psi'(x)=\varphi((1-s(\psi))x(1-s(\psi)))$.

2. Taking $\psi=\omega$, you have $$ \omega(x)=\langle \pi_\omega(x)\,\Omega,\Omega\rangle,\ \text{ for some } \Omega\in\mathfrak H_\omega. $$ Let $\xi=u\Omega$. Then $$ \omega(x)=\langle u^*\pi_\omega(x)u\Omega,\Omega\rangle =\langle \pi_\varphi(x)u\Omega,u\Omega\rangle =\langle \pi_\varphi(x)\xi,\xi\rangle, $$ where $\xi=u\Omega$.