Here is a quotation of a book:
Let $S(A)$ denote the state space of a C*-algebra $A$ and $M\subset S(A)$ denote a weak-$*$ closed convex set. Assume there is a state $\psi$ which does not belong to $M$. By the Hahn-Banach Theorem we can find $a\in A$ and a real number $t$ such that $$Re(\phi(a))<t<Re\psi(a).$$ for every $\phi\in M.$
In my view, if $M$ is a closed convex set, then, from Hahn-Banach Theorem, we have a continuous linear functional $\rho$ on $M$ such that $Re(\rho(\phi))<t\leq Re(\rho(\psi))$. But, how to find such a $a\in A$ in the quotation?