Stone-weierstrass Theorems for the Strict Topology

1. Let X be a locally compact Hausdorff space, E a (real) locally convex, complete, linear topological space, and (C*(X, E), ß) the locally convex linear space of all bounded continuous functions on X to E topologized with the strict topology ß. When E is the real numbers we denote C*(X, E) by C*(X) as usual. When E is not the real numbers, C*(X, E) is not in general an algebra, but it is a module under multiplication by functions in C*(X). This paper considers a Stone-Weierstrass theorem for (C*(X), ß), a generalization of the Stone-Weierstrass theorem for (C*(X, E), ß), and some of the immediate consequences of these theorems. In the second case (when E is arbitrary) we replace the question of when a subalgebra generated by a subset S of C*(X) is strictly dense in C*(X) by the corresponding question for a submodule generated by a subset S of the C*(X)-module C*(X, E). In what follows the symbols Co(X, E) and Coo(X, E) will denote the subspaces of C*(X, E) consisting respectively of the set of all functions on I to £ which vanish at infinity, and the set of all functions on X to £ with compact support. Recall that the strict topology is defined as follows [2 ] : Definition. The strict topology (ß) is the locally convex topology defined on C*(X, E) by the seminorms