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Consider the $C^*$-algebra $A := C(X)$, where $X$ is a compact Hausdorff space. Denote by $S(A)$ the state space of $A$. In the weak$^*$ topology this is a convex compact set such that $S(A)$ is the closed convex hull of its extreme points. By using the Riesz-Kakutani representation theorem it is easy to see that the pure states (extreme points of $S(A)$) are exactly the point evaluations.

Is there an adhoc proof of this fact ?

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  • $\begingroup$ Which fact do you mean? That the pure states on $C(X)$ are exactly the point evaluations? $\endgroup$ – MaoWao Oct 7 '16 at 10:05
  • $\begingroup$ Yes, you are right. $\endgroup$ – user42761 Oct 7 '16 at 10:41
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    $\begingroup$ If $\varphi$ is a pure state, then the corresponding representation $\pi_{\varphi}$ must be one-dimensional (since $C(X)$ is commutative). Hence, $\ker(\pi_{\varphi})$ is a maximal ideal, which must coincide with the set of functions that vanish at a point. Hence, $\varphi$ must be the evaluation at that point. $\endgroup$ – Prahlad Vaidyanathan Oct 7 '16 at 12:27

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