A character extends to a pure state

Write for the state space of . The character is a positive normalized functional of norm one on . Apply the norm-preserving Hahn–Banach extension in A norm-preserving Hahn–Banach extension is an ambient state to the underlying complex subspace and this functional. Obtain with and . Since , . The -algebra criterion and implies positivity. Thus the set

of state extensions is nonempty.

Give the continuous dual the weak-star topology. Each condition is closed because evaluation at a fixed is continuous, and the state conditions are weak--closed. States have norm at most one, so the closed set lies in the weak-star compact dual unit ball and is compact. It is convex because positivity, normalization and all restriction equations are preserved by real convex combinations. These are the compactness and nonemptiness facts in Compactness, nonemptiness and the face property.

The character is pure on by A character is a pure state. Explicitly, if with states on and , for an arbitrary put . Then , while both and are nonnegative. The convex equality forces both to vanish. Cauchy–Schwarz gives and the corresponding inequality for . Thus , so normalization gives for every .

Let and satisfy

Their restrictions to are states, and restriction of this equation gives

Step 3 implies , so . This is exactly the face property proved in Compactness, nonemptiness and the face property.

The extreme-point existence theorem for a nonempty compact convex set in the real locally convex weak-star dual gives a state . Since is a face of , the same is extreme in . Its membership in gives positivity, and for every ; its extremality gives purity. These are all the required conclusions for one continuous complex-linear functional.