Representation theorems for closed ideals of CB (X)

Abstract

Let CB (X) be the Banach algebra of all continuous bounded ℂ-valued functions on a completely regular Hausdorff space X equipped with the supremum norm. We provide representation theorems for arbitrary closed ideals of CB (X). Indeed, for a closed ideal H of CB (X) we associate a subspace of the Stone-Čech compactification of X such that H and are isometrically isomorphic. The space , which is locally compact, is uniquely determined by this property. As a consequence of our representation theorem we explicitly construct a dense subideal in H which is isometrically isomorphic with . We further associate an open subset U of X and a (set theoretic) ideal in X to H such that H is isometrically isomorphic with . We prove that there is the smallest ideal in X such that . From this we derive several properties of closed ideals of CB (X). For example we show that the set of all closed ideals of a commutative C*-algebra (with or without unit) forms a distributive lattice. As an another example we show that for every completely regular Hausdorff space X and every ideal in X, the completion of the normed ideal of CB (X) is the space .