Let $\omega$ be a $(\tau, \beta)$- KMS state of the $C^*$- dynamical system $(\mathcal{U}, \tau)$ with $\beta \in \mathbb{R}-\{0\}$. What is mean by the normal extension of $\omega$ to the weak closure $\mathcal{M}_\omega=\pi_\omega(\mathcal{U})''$ of $\mathcal{U}$ in the GNS representation $(\mathcal{U}_\omega, \pi_\omega, \Omega_\omega)$.
How is it defined? Can I get a reference for this. I am stuck on the corollary 5.3.4 in the book Operator algebras and Quantum Statistical Mechanics 2 by O Brattelli and D W Robinson.
What you do is simply define $$ \hat\omega(x)=\langle x\,\Omega_\omega,\Omega_\omega\rangle. $$ This is a normal state, and for any $a\in\mathcal U$ you have $$ \hat\omega(\pi_\omega(a))=\langle \pi_\omega(a)\,\Omega_\omega,\Omega_\omega\rangle=\omega(a). $$