Decomposable compact convex sets and peak sets for function spaces.

Abstract

Geometric conditions are known under which a closed face of a compact convex set is a peak set with respect to the space of continuous affine (real-valued) functions. The purpose of this note is to give an application of this 'abstract-geometric set-up to the problem of finding peak sets (or points) in a compact Hausdorff space with respect to a closed subspace of continuous complex-valued functions. In this fashion we obtain the strong hull criteria of Curtis and Fig&-Talamanca and in particular the Bishop peak point theorem for function algebras. Let X be a compact convex subset of a Hausdorff locally convex space and let A (X) denote the space of continuous real-valued affine functions on X. Then A (X) is a Banach space under the supremum norm. We make the usual identification of X with the set of positive normalized functionals in A (X)* with the weak* topology. If F is a closed face of X we say X is decomposable at F underf (2 ) iffEA (X) ** such thatf is identically zero on the weak* closed linear span of F and