# On examples of subspaces of $C(X)$ for which state spaces are Choquet Simplices

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 $$K$$.

• Yes, but I need an example with Real scalar. The space $C(X)$ I considered with Real scalars. – Tanmoy Paul Jun 16 '20 at 6:58