I'm looking for an example of a non-unital $C^*$-algebra $A$ whose set of states $S(A)$ is not compact (in the weak* topology, of course).
I think $K(H)$, the compact operators over a Hilbert space $H$, is a good place to start. However, I don't see why $S(K(H))$ is not compact. Any insight would be appreciated.
The first natural thing to look at would be continuous functions vanishing at infinity on a locally compact non-compact Hausdorff space. Can you find a closed non-compact subset of the state space of this algebra?