Stone–Weierstrass Theorems in C*(X) Original Research Article

In this paper we consider some conditions for a given function f∈C*(X) to belong to the uniform closure of a subset W⊂C*(X). We start with a general theorem which admits a sensible improvement when W, or more generally its uniform closure cl(W), is a lattice. Also, we obtain an approximation result when W or cl(W) is a cone. From this result we can derive one by Blasco and Moltó for linear subspaces and one by Garrido and Montalvo for semi-affine lattices. Finally, using “multipliers,” we extend to C*(X) some other known results for the compact case, such as the Nachbin theorem about the uniform closure of an A-module.