Unique decomposition property and extreme points (Q734572)

Let \(X\) be a compact Hausdorff space and \(H\) be a space of continuous real valued functions on \(X\) which separates points and includes the set \(C\) of all constant functions on \(X\). We denote by \(\text{Ch}_H(X)\) the Choquet boundary of \(H\) on \(X\) and by \((H/C)^*\) the dual space of the quotient space \(H/C\). We can identify \((H/C)^*\) with the subset \(\{\psi\in H^*: \psi(C)= 0\}\) of \(H^*\). We define \(B^+_{H^*}= \{\psi\in H^*: pis\geq 0\), \(\|\psi\|\leq 1\}\) and write \(\text{ext}(M)\) for the set of extreme points of the convex set \(M\). For each \(x\in X\), we define the functional \(\phi_x(f)= f(x)\) for all \(f\in X\). A result of \textit{J. J. Font} and \textit{M. Sanchis} [Rocky Mt. J. Math. 34, No. 4, 1325--1331 (2004; Zbl 1073.46016)] says that \[ \text{ext}(B^+_{(H/C)^*}\subset \{\phi_x- \phi_y: x,y\in \text{Ch}_H(X),\quad x\neq y\}. \] The object of the present paper is to show that this set inclusion becomes an equality if and only if a further condition on \(H\), called the unique decomposition property, is satisfied.