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Let $C(X)$ be the Banach space of all Real valued continuous functions on a compact Hausdorff space $X$. What are examples of uniformly closed subspace $\mathcal{A}$ of $C(X)$ such that $\mathcal{A}$ separates points, containing constants and the state space of $\mathcal{A}$ is a Choquet Simplex.

The state space of $\mathcal{A}$ viz. $S_{\mathcal{A}}$ is defined as $\{\Lambda\in\mathcal{A}^*:\|\Lambda\|=1 ~\mbox{and}~ \Lambda (1)=1\}$. By a Choquet Simplex we mean a compact convex subset $K$ of a locally convex topological vector space $E$ such that for each $p\in K$, there exists a unique measure $\lambda$ on $K$ such that $f(p)=\int_K f(t)d \lambda (t)$, $\forall~ f\in E^*$. When $K$ is metrisable then $\lambda$ can be assumed to satisfy $S(\lambda)\subseteq ext (K)$ but for non metrisable case $S(\lambda)\subseteq \overline{ext(K)}$, in some sense these measures are 'maximal'. $ext(K)$ represents the set of all extreme points of $K$.

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On page 148 of Convexity Theory and its Applications in Functional Analysis, L. Asimow and A.J. Ellis say that every Dirichlet algebra (an algebra where the real parts of its elements are dense in the space of real-valued continuous functions) gives rise to a simplicial state space. The disc algebra is an example of a Dirichlet algebra.

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  • $\begingroup$ Yes, but I need an example with Real scalar. The space $C(X)$ I considered with Real scalars. $\endgroup$ – Tanmoy Paul Jun 16 at 6:58

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