Let $A$ be a unital $C^{\ast}$-algebra and let $P(A)$ be a space of pure states on $A$ (a state $\omega$ is called pure it is an extreme point in space of states).

a) Is the space $P(A)$ compact with respect to the weak-$\ast$ topology? In the commutative case it appers that $P(A)$ is the space of nonzero linear characters and this space is compact, is this also true in the noncommutative case?

b) In concrete situation, $A=B(H)$, the space of bounded linear operators on the Hilbert space is it true that $P(A)$ is extremely disconnected? I would be grateful if anyone could help.