TITLE: Existence of faithful state in $C^\ast$-algebras
QUESTION [5 upvotes]: Why does there always exist a faithful state in a separable $C^\ast$-algebra?

REPLY [6 votes]: Well, since the unit ball in the dual space is weak-$^*$-separable, there is a separating sequence of states $\omega_n$, then just sum them up $\omega=\sum 2^{-n}\omega_n$ and check that you get a faithful state.
Credits goes to t.b.