Stone–Weierstrass theorems for Riesz ideals of continuous functions

Notions of convergence and continuity specifically adapted to Riesz ideals $$\mathscr {I}$$ of the space of continuous real-valued functions on a Lindelöf locally compact Hausdorff space are given, and used to prove Stone–Weierstrass-type theorems for $$\mathscr {I}$$ . As applications, sufficient conditions are discussed that guarantee that various types of positive linear maps on $$\mathscr {I}$$ are uniquely determined by their restriction to various point-separating subsets of $$\mathscr {I}$$ . A very special case of this is the characterization of the strong determinacy of moment problems, which is rederived here in a rather general setting and without making use of spectral theory.