Smooth Functionals on Lattices of Continuous Functions

Abstract

We introduce the notions of tight, τ-smooth and σ-smooth linear functionals on locally convex vector lattices as a generalization of the tight, τ-smooth and σ-smooth linear functionals, respectively, on the space Cb(X) of all bounded, continuous, real-valued functions on the completely regular space X. It is shown that many of the order-theoretical properties of these functionals on Cb(X) are still shared by their respective generalizations. The functionals of different smoothness type are used to introduce different types of function lattices, Cb(X) with its different strict topologies being a particular example. Among the results which are still true in this more general setting is the Stone-Veierstrass theorem proved for quasi-M-spaces and generalizing the corresponding theorem for Cb(X) with the substrict topology. Also the various ideal theoretical properties of Dini- and σ-Dini-spaces shed some new light upon Cb(X) with the strict and superstrict topology, respectively.