Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.