On Šilov boundaries for subspaces of continuous functions

In this paper we prove that if A is a strongly separating linear subspace of C0(X), that is, for every χ, y, X there exists A such that (χ) ≠ (y), then the Šilov boundary for A exists 0605 1069 V 3 and is the closure of the Choquet boundary for A. addition, we assume that A is a closed subalgebra, then we provide a proof of the following: the strong boundary points for A (peak points when X satisfies the first axiom of countability) are dense in the Šilov boundary. Indeed they are a boundary for A. Our proof does not depend on the analogous results for separating closed subalgebras of C(X) (X compact) which contain the constant functions, that is, uniform algebras.