Pure state on a C-star algebra

A pure state on a C-star algebra is an extreme point of its state space. For a nonzero unital C-star algebra, the state space is a nonempty compact convex set in the weak-star topology, so the Krein-Milman theorem guarantees pure states.

Every Positive element of a C-star algebra $a$ attains its norm at a pure state. Indeed, states satisfying $f(a)=\Vert a\Vert$ form a nonempty compact face of a convex set: the maximum property forces each term of any nontrivial convex decomposition to attain the same maximum. An extreme point of this face is an extreme point of the whole state space. The argument uses the unital compactness statement, rather than asserting compactness for a general nonunital algebra.