Is the set of states of an infinite dimensional unital $C^*$-algebra non-compact in the norm topology?

Question

Let $\mathfrak{A}$ be an infinite dimensional unital $C^*$-algebra. I know that the ball $B_{1}$ of radius one in the dual space $\mathfrak{A}^*$ is non-compact in the norm topology, but what about the set of states $E_\mathfrak{A}\subset B_{1}$ on $\mathfrak{A}$?

Answer 1

We can write each linear functional in $B_1$ as a linear combination of four states, with all coefficients in the closed unit disk. It follows that if $E_{\mathfrak A}$ is compact, so is $B_1$.