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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'.

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 $K$.