I'm interested in knowing the requirements for a $*$-representation, $\pi_{\omega}$, of a C*-algebra, $\mathbb{C}(\mathcal{G})$, (or equivalently the requirements for the unitary representation, $U_{\mathcal{G}}$, corresponding to $\pi_{\omega}$) to have a unique cyclic and separating vector $\xi$.
P.S. I'm a physicist and I'm new to this field so I would appreciate a more detailed answer. Thank you so much in advance!!
