Simplicial function spaces and simplexes (Q1058105)

Let H be a function space on a compact metrizable space X. H is called simplicial if the state space S(H) of H is a Choquet simplex. H is called weakly simplicial if, for every point \(x\in X\), there exists exactly one H-representing measure for x which is supported by the Choquet boundary \(Ch_ H(X)\). It is proved that H is weakly simplicial if and only if \(\hat H\) is simplicial. Here, \(\hat H\) is the space of all H-affine functions on X, i.e. the space of all \(f\in C(X)\) having the same representing measures as H.