Proposition 34.10.4.label Let $A$ be a unital $C^{*}$-algebra, then $S(A)$ is a compact convex set, and $S(A)$ is the weak*-closed convex hull of $P(A)$.
Proof. Since the evaluation map is weak* continuous and
\[S(A) = \bracs{\phi \in A^*|\dpn{1, \phi}{A} = 1}\cap \bigcap_{\substack{x \in A \\ x \ge 0}}\bracs{\phi \in A^*|\dpn{x, \phi}{A} \ge 0}\]
the state space is an intersection of convex and weak*-closed sets, so it is closed and convex.
By Theorem 34.9.2, $S(A) \subset \ol{B_{A^*}(0, 1)}$, which is weak* compact by the Banach-Alaoglu Theorem. Therefore $S(A)$ is compact by Proposition 5.16.3.
By the Krein-Milman Theorem, $S(A)$ is the weak*-closed convex hull of $P(A)$.$\square$
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