A general approach to the notion of Silov boundary (Q1113060)

Suppose that X is a locally convex complex space and that K is a compact subset of X such that the closed convex hull \(\overline{conv K}\) of K is compact. It is shown that the closure of extreme points of \(\overline{conv K}\) is the smallest closed subset of K on which the supremum of the modulus of every functional f from the dual of X is equal to the supremum of the modulus of f on the whole of K. The above is shown under the assumption that K lies on some hyperplane away from the origin. The result is applied to get an immediate existence proof of the Shilov boundary of a commutative unital Banach algebra.