Intermediate algebras between C∗(X) and C(X) as rings of fractions of C∗(X)

Let C(X) be the algebra of all K-valued continuous functions on a topological space X (with K = R or K = C) and C∗(X) the subalgebra of bounded functions. This paper deals with subalgebras of C(X) containing C∗(X). We prove that these subalgebras are exactly the rings of fractions of C∗(X) with respect to multiplicatively closed subsets whose members are units of C(X). As rings of fractions these intermediate algebras inherit some algebraic properties from C∗(X) but, in general, they are neither isomorphic to any C(T) nor even closed under composition. We characterize these two kinds of intermediate algebras by means of algebraic properties of the corresponding multiplicatively closed subsets, and we show that the intermediate algebras isomorphic to some C(T) are exactly those that are closed under inversion.