Relative Choquet boundaries of restriction algebras

Abstract

Given a vector subspace of a topological algebra, along with a subset of its spectrum, we characterize the Choquet boundary of the restriction of the Gel'fand transform algebra on the previous subset, relative to the ``Gel'fand transform'' of the given subspace, restricted on the same subset. This is accomplished, by employing the geometric hull of the above subset and the respective strong points of the algebra, relative to the subspace involved. We also compare the relative Choquet boundaries for different subsets of the spectrum, via appropriate identifications, so that the closed convex hulls of the spectra, of the algebra and of its restricted Gel'fand transform, coincide. Finally, the Choquet boundaries of the latter algebras are also compared, in the case the above subset is a weakly peak set.