Homomorphisms on Function Lattices

Abstract

In this paper we study real lattice homomorphisms on a unital vector lattice \({{\cal L}} \subset {\cal C}(X)\), where X is a completely regular space. We stress on topological properties of its structure spaces and on its representation as point evaluations. These results are applied to the lattice \({{\cal L}} = {\rm Lip} (X)\) of real Lipschitz functions on a metric space. Using the automatic continuity of lattice homomorphisms with respect to the Lipschitz norm, we are able to derive a Banach-Stone theorem in this context. Namely, it is proved that the unital vector lattice structure of Lip (X) characterizes the Lipschitz structure of the complete metric space X. In the case \({{\cal L}} = {\rm Lip}^{\ast} (X)\) of bounded Lipschitz functions, an analogous result is obtained in the class of complete quasiconvex metric spaces.