The density of peak points in the Shilov boundary of a Banach function algebra

Abstract:

H. G. Dales has proved in [1] that if $A$ is a Banach function algebra on a compact metrizable space $X$, then ${\bar S_0}(A,X) = \Gamma (A,X)$, where ${S_0}(A,X)$ is the set of peak points of $A$ (w.r.t. $X$) and $\Gamma (A,X)$ is the Shilov boundary of $A$ (w.r.t. $X$). Here, by considering the relation between peak sets and peak points of a Banach function algebra $A$ and its uniform closure $\bar A$, we present an elementary and constructive proof of the density of peak points in the Shilov boundary.