A STONE-WEIERSTRASS-TYPE THEOREM FOR TRUNCATED VECTOR LATTICES OF FUNCTIONS

Abstract

A nonempty set S of real-valued functions is said to be truncated if it contains with any function f its meet with the constant function 1. Very recently, the first named author and Hajji obtained a Stone-Weierstrass-type theorem for a truncated vector sublattice L of $C_{0}\left( X\right)$, where X is a locally compact Hausdorff space. Relying heavily on the multiplicative version of the classical Stone-Weierstrass theorem, they proved that if L separates the points of X and vanishes nowhere on X, then L is uniformly dense in $C_{0}\left( X\right)$. In this paper, we shall provide a new, intrinsic, and more natural proof of this result. We shall then apply the result to prove that any truncated vector lattice of bounded real-valued functions is essentially a $C_{0}\left( X\right)$-type space for some locally compact Hausdorff space X. This yields, in particular, that for any arbitrary topological space Y, a locally compact Hausdorff space X can be found such that $C_{0}\left( Y\right)$ and $C_{0}\left( X\right)$ are essentially the same, eliminating any reason for considering Banach lattices of continuous functions vanishing at infinity on other than locally compact Hausdorff spaces.