# Affine Homeomorphism between a compact set K and the state space on A(K)

Let $V$ be a locally convex space, and let $K$ be compact set in $V$. Define $A(K)\subset C(K)$ as $A(K)=\{\phi:K\rightarrow \mathbb{C}\; |\; \phi\; \text{is continuous and affine}\}$. Then we know that $A(K)$ is a function system. And hence we can define its state space as $S(A(K))=\{f:A(K)\rightarrow\mathbb{C}\;|\; f \;\text{is positive and } f(1)=1\}$. Then again from basic $C^*$ theory we know that $S(A(K))$ is weak* compact. The question that is bothering me is this. I need to prove that $K$ and $S(A(K))$ are affinely homeomorphic. The map I have defined is $x\mapsto \hat{x}$, where $\hat{x}$ is the usual evaluation map. I have shown almost everything except that this map is an onto map. How do I prove this part? Any reference or hint will be appreciated. Thanks.

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