Texts: Let A be a C*-algebra with an identity element, and let SA denote its state space. It is a fact—you need not prove this—that SA is non-empty.
(a) It is stated in Proposition VIII.5.15 on page 252 of Conway's book that SA is a weak*-compact and convex subset of the dual space A* of A, but the proof of this statement is only sketched. Without merely quoting statements from that sketch, give a detailed and convincing proof of the following facts:
(i) SA is a weak*-closed subset of A*.
(ii) SA is a weak*-compact subset of A*.
(iii) SA is a convex subset of A*.
A state on A is called a pure state on A when it is an extremal point of SA.
(b) Show that there exists a pure state on A.
(c) It is a fact—you need not prove this—that SA separates the points of A. Show that the set of pure states on A also has this property. That is, show that a = 0 whenever a ∈ A is such that f(a) = 0 for every pure state f on A.