Markov operators, peak points, and Choquet points

Abstract:

We deal with conditions on a closed subspace $L$ of $C(X)$ under which weak peak points are equivalent with Choquet points. If $L$ satisfies a strengthened form of this equivalence (’weak peak sets = Choquet sets’—definitions are given below), then for any image of $L$ under a Markov projection, weak peak points and Choquet points are equivalent. Conditions under which $L$ satisfies the strengthened equivalence include (i) $L$ interpolates its Shilov boundary, and (ii) $L$ is the space of fixed points of a Markov operator $T$ on $C(X)$, where $T$ is in an appropriate sense concentrated on the Shilov boundary of $L$.