REPRESENTATION THEOREMS FOR CLOSED IDEALS OF CB(X)

Let C-B (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 C-B (X). Indeed, for a closed ideal H of C-B (X) we associate a subspace sp (H) of the Stone-Cech compactification of X such that H and C-0 (sp (H)) are isometrically isomorphic. The space sp (H), 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 C-00(sp (H)). We further associate an open subset U of X and a (set theoretic) ideal J in X to H such that H is isometrically isomorphic with C-0(J)(U). We prove that there is the smallest ideal J(H) in X such that H = C-0(JH) (X). From this we derive several properties of closed ideals of C-B(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 J in X, the completion of the normed ideal C-00(J) (X) of C-B (X) is the space C-0(J) (X).