Consider a separable $C^*$ algebra $\mathcal A$. The space of states is also separable in the weak* topology, let $S$ be a countable dense subset.

Denoting with $H_\omega$ the GNS representation of a state $\omega$ we retrieve a representation of $\mathcal A$ on the Hilbert space $H(S)=\bigoplus_{\omega\in S}H_\omega$. This representation is isometric and $H$ is separable.

In the context of quantum mechanics we have built a candidate for the physical Hilbert space just by knowing an algebra of observables. This construction however depends on how we chose our set $S$. My question is:

Are the representations of $\mathcal A$ on $H(S)$ and $H(S')$ unitarily equivalent for any two dense countable subsets $S,S'$ of the state space of $\mathcal A$?