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I posted this question a few days back at math.SE but could not get help. So I was kind of forced to ask here. Please excuse me if this question is not suitable here.

Let $V$ be a locally convex space, and let $K$ be convex 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

$$S(A(K))=\{f:A(K)\rightarrow\mathbb{C}\;|\; f \;\text{is positive and } f(1)=1\}$$ We also 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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  • $\begingroup$ I assume that you mean $K$ is convex and compact. Otherwise convex combinations of states $\hat{x}$ are not necessarily of the form $\hat{y}$ with $y\in K$. For example, consider the unit interval $I$ in $\mathbb{R}$. Then $A(I)=A(\\{0,1\\})$. $\endgroup$
    – Steven Deprez
    Commented Mar 23, 2012 at 10:00
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    $\begingroup$ Does Aaron's answer below not suffice? $\endgroup$
    – Yemon Choi
    Commented Jun 20, 2012 at 4:31

1 Answer 1

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I believe that this is Theorem 7.1 in Ken Goodearl's book "Partially ordered abelian groups with interpolation."

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