Bishop's Generalized Stone-weierstrass Theorem for the Strict Topology

1. Let X be a locally compact Hausdorff space, C(X)ß the locally convex topological vector space obtained from all bounded complex continuous functions on X by employing the strict topology [2]. The present note is devoted to a version of Bishop's generalized StoneWeierstrass theorem [l] applicable to certain subspaces of C(X)ß-, essentially it is a footnote to an earlier paper [4], in which a modification of de Branges' proof of the Stone-Weierstrass theorem [3] was used to obtain Bishop's theorem. Insofar as possible the notation will be that of [4]. The version of Bishop's theorem we shall write down was motivated by, and has application to, the spectral theory of bounded continuous functions on locally compact abelian groups, where the strict topology enjoys a useful rôle [5]. However, our applications to spectral theory amount at best to new proofs of known results. Recall that the strict topology has a base of neighborhoods of 0EC(X)ß of the form