States on a $C^*$-algebra

Question

I know that if $A$ is a non-zero and unital $C^*$-algebra then $S(A)$ (the set of states on it) is weak${}^*$ compact.

My problem is:

Does the same hold if $A$ is not unital?

Answer 1

Consider $A=C_0(\mathbb R)$, and let $\varphi_n\in S(A)$ be given by $$\varphi_n(g)=g(n).$$ Then $\varphi_n\to0$ pointwise, and so $S(A)$ is not closed.

With a similar idea: you can take $A=K(\ell^2(\mathbb N))$ and, with $\{e_n\}$ the canonical basis, $$ \varphi_n(x)=\langle xe_n,e_n\rangle. $$