TITLE: The space of pure states
QUESTION [5 upvotes]: 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.

REPLY [5 votes]: a) The set of pure sates may not be *-closed in the dual space of $A$, see for example a paper of J.Glimm: http://www.ams.org/journals/tran/1960-095-02/S0002-9947-1960-0112057-5/ So of course it may be not compact.
Maybe it is reasonable to consider the spectrum of $A$ insead, i.e. the space $\hat A$ of all classes of irreducible representations of $A$. Look at Dixmier "C*-algebras and their representations", chapter 3. In particular, 3.1.8 tells you that if $A$ is unital then $\hat A$ and the space of primitive ideals are both compact but maybe not Hausdorff.
b) For $A=B(H)$, he set $P(A)$ is much bigger than simply the set of states of tpe $f_x(a)=(ax,x)$ for $x\in H$, $a\in A$. But the set of these $f_x$ is arc connected, because the unit sphere in $H$ is arc connected. So $P(A)$ cannot be totally disconnected.
[Hope I didn't write anything silly at this late hour]