Let $A$ be a unital $C^*$-algebra, and let $S(A)$ denote the set of states on $A$, that is, the set of positive linear functionals $\rho:A\to\mathbb C$ satisfying $\rho(1_A)=1$. Then $S(A)$ is a convex weak* compact subset of $A^*$, where $A^*$ is equipped with the [weak* topology](/page/Weak*%20Topology) $\sigma(A^*,A)$.