My question is whether there exists a unique cyclic and separating state on a C$\ast$-algebra giving rise to a specific representation (in other words is it possible to reverse the GNS construction)? If there is, then how can we find it ? If not, are there any specific extra constraints that we should impose on the state to be unique?
Since there is a bijection correspondence between non-degenerate representation $\pi$ of a $\ast$-algebra $\mathbb{C}\mathcal{G}$ and unitary representation $U$ of $\mathcal{G}$, I'm ultimately interested in finding a unique cyclic and separating state,$\omega_U$, over $\mathbb{C}\mathcal{G}$ given the unitary representation $U$ of $\mathcal{G}$. To be more specific I want to start with a unitary representation of a Lie group (SU(N) for instance) and find the corresponding cyclic and separating state to that representation on the group C$\ast$-algebra.