Weakly prime sets for function spaces

A function space A on a compact Hausdorff space X is a closed subspace of the space C(X) of all continuous, complex-valued functions on X separating points and containing constants. If A is an algebra, it is called a function algebra. The Bishop and Silov decompositions play an important role in characterizing function algebras. Later on these decompositions were studied for function spaces [6]. Ellis [3] defined and studied these decompositions for the spaces of affine functions on a compact convex set. For a function algebra, certain decompositions finer than the Bishop and Silov decompositions have been defined and studied [6]. One such decomposition of weakly prime sets, was defined and discussed by Ellis [4] for function algebras as well as for spaces of affine functions. Here we generalize this concept for a function space, study its properties and show that it coincides with the corresponding definitions of Ellis. We also give examples of function spaces whose family of maximal weakly prime sets differ from the corresponding families of its induced algebras.