An application of Hahn-Banach theorem in the context of $C^{\ast}$-algebra.

Let $A$ be a $C^{\ast}$-algebra and let $S(A)$ be the set of all states on $A.$ Define a set $M$ as follows $:$ $$M : = \left \{t \omega_1 - (1 - t) \omega_2\ \big |\ \omega_1,\omega_2 \in S(A),\ t \in [0,1] \right \}.$$ If there exixts a hermitian functional $f_0$ on $A$ with $\left \|f_0 \right \| \leq 1$ such that $f_0 \notin M$ then using Hahn-Banach theorem show that there exists $a \in A$ and $t \in \mathbb R$ such that $\text {Re}\ f_0 (a) \gt t$ and $\text {Re}\ f (a) \leq t$ for all $f \in M.$

I am getting stuck here. I don't know which version of Hahn-Banach theorem would give me the required conclusion. A small hint would be a boon for me at this stage.

Thanks for your time.

Consider the geometric Hahn-Banach theorem. By your assumption we have $$M \cap \{f_0\}= \emptyset.$$ Note that $M$ is convex (it is the convex hull of $S(A)\cup (-S(A))$) and weak$^*$-compact (as the compact image under a continuous function), so by the Hahn-Banach separation theorem, there exists a weak$^*$-continuous functional $$\omega: A^* \to \mathbb{C}$$ and a real number $t \in \mathbb{R}$ such that $$\Re \omega(\gamma) \le \lambda < \Re \omega(f_0), \quad \gamma \in M.$$ Next, observe that every weak$^*$-continuous function $\omega: A^*\to \mathbb{C}$ is necessarily of the form $$\omega = \operatorname{ev}_a$$ for some $a \in A$, so we obtain $$\Re \gamma(a) \le t < \Re f_0(a), \quad \gamma \in M.$$