State space is weak* compact

Question

I'm trying to convince myself that the state space $S(A)$ of a unital $C^*$-algebra is weak* compact. I've proven that $S(A)$ is convex, and I feel that this should allow me to conclude weak* compactness. However, after awhile, I don't see how this could be the case. Can someone point me in the right direction?

Answer 1

Use the Banach–Alaoglu theorem. Now you only need to prove that $S(A)$ is weakly* closed.

Elaboration: The weak* topology is, by definition, the weakest topology on $A^*$ for which every bounded linear functional of the form $\psi\mapsto\psi(a)$, with $a\in A$, is continuous. In particular, applying this to the unit element $e$, we conclude that $\{\,\psi\in A^*\colon \psi(e)=1\,\}$ is weakly* closed. The state space is just the intersection of this hyperplane with the unit ball of $A^*$, which is also weakly* closed (which follows from compactness, but more easily from Hahn–Banach).