Let $A\subseteq B \subseteq B(H)$ be an inclusion of $C^\*$-algebras where $H$ is some Hilbert space. We have the following conditions:


* B is a von Neumann algebra with $A'' = B$.
* The inclusion $A \subseteq B(H)$ is non-degenerate.
* $B$ contains $\operatorname{id}\_H$, but $A$ does not.


Further, let $\omega \in B^\*$ be a normal state and let $(H\_\omega, \pi\_\omega, \xi\_\omega)$ the associated GNS-triplet. If $\{u\_\lambda\}\_{\lambda\in \Lambda}$ is an approximate unit for $A$, is it true that
$$\lim\_{\lambda \in \Lambda}\|\pi\_\omega(u\_\lambda)\xi\_\omega-\xi\_\omega\| = 0?$$


I know that the representation $\pi\_\omega$ is a normal $\*$-morphism. Maybe that's useful here.



Show convergence of net associated to GNS-triplet associated to state on a $C^*$-algebra
We need the following fact: if a net of positive operators $(a\_{\lambda})$ is increasing and bounded then it converges in the strong operator topology. It is not difficult to check that it converges in the weak operator topology to some element $a$. Therefore $(a-a\_{\lambda})$ is a bounded net of positive operators that converges to zero. It follows that the net formed by the square roots $(\sqrt{a-a\_{\lambda}})$ converges to zero in the strong operator topology. Recall now that multiplication is continuous with respect to the strong operator topology on bounded subsets. As the net $(\sqrt{a-a\_{\lambda}})$ is bounded, the net formed by squares, i.e. $(a-a\_{\lambda})$ converges to zero in the strong operator topology as well. It means that $(a\_{\lambda})$ converges to $a$ in the strong operator topology.


By your assumptions, the approximate unit $(u\_{\lambda})$ converges to the identity in the weak\* topology. From normality of $\pi\_{\omega}$ it follows that $(\pi\_{\omega}(u\_{\lambda}))$ converges to the unit in the weak\* topology. But $(\pi\_{\omega}(u\_{\lambda}))$ is a bounded, increasing net, thus it converges in the strong operator topology. The limits in the two topologies have to agree, therefore $(\pi\_{\omega}(u\_{\lambda}))$ converges to the unit in the strong operator topology, in particular $\lim\_{\lambda \in \Lambda} \|\pi\_{\omega}(u\_{\lambda})\xi\_{\omega} - \xi\_{\omega}\|=0$.


