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I have two problems to ask. Let $A$ be a function algebra of $C(K)$. $t\in K$ is said to be a peak point of $A$ if $\exists~f\in A$ s.t. $|f(t)|=\|f\|$ and $|f(s)|<|f(t)|$ for any $s\neq t$. Suppos …

Let $M^+(D)$ be the space of all positive measures on a closed convex subset $D$ of a locally convex topological vector space $E$. Two measure $\mu, \nu\in M^+(D)$ one can define a partial ordering $\ …

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 $\mathcal{A}$ …

Solution (1). Let $A=\{f\in A(\mathbb{D}):f(0)=f(1)\}$. It follows from the Maximum Modulus Principle that the Choquet boundary $Ch(A)$ does not contain $1$, as $B(A)$ and the set of peak points are …