On completely additive state in von Neumann algebra

Question

It is unclear to me in a universal normal representation of von Neumann algebra $M$ say, how any completely additive state looks like vector state. I am reading the lemma 7.1.6 in Kadison Ringrose and I am not able to understand the proof. If anybody gives the universal normal representation idea of von Neumann algebra what it exactly means by giving example I will be really helpful to learn.

Answer 1

If you read before Lemma 7.1.6, near the bottom of page 458, the normal universal representation is defined to be the direct sum of the GNS representations of the additive states.

That is, $\Phi_n=\bigoplus_\eta \pi_\eta$ where $\eta$ runs over the completely additive states, and $\pi_\eta$ is the GNS representation. Explicitly, $$ \omega(a)=\langle \pi_\omega (a) \Omega_\omega,\Omega_\omega\rangle=\langle \Phi_n(a)x,x\rangle, $$ where $x=(x_\eta)$ is given by $x_\omega=\Omega_\omega$ and $x_\eta=0$ if $\eta\ne\omega$.