Boundaries and peak points in topological algebras

Abstract

We examine the relation of the set of peak points with several boundaries defined in the spectrum of a given topological algebra relative to a subspace of it. In this respect, we show that the peak points are contained in the Choquet boundary and, under suitable conditions, are dense in the Šilov boundary. Furthermore, the set of points at issue coincide with the Bishop boundary if, and only if, it constitutes a weakly boundary set. On the other hand, in appropriate topological algebras, the Bishop, Choquet and strong boundaries coincide with the peak points, so that they are dense in the Šilov boundary. Finally there are topological algebras for which all the above boundaries and points remain invariant, under restriction of the Gel'fand transform algebras to subsets of the spectra of the topological algebras involved, containing the Šilov boundary.