= Pure state on a C-star algebra
A <pure state on a C-star algebra> is an <extreme point> of its <state on a C-star algebra>[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 $f(a)=\|a\|$ 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.
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