On vector lattices of continuous functions in locally compact spaces

Abstract

Definitions and notations. Let T be a Tihonov space, C(T) = { f l f : T~R a n d f i s a continuous function}, C*(T)= {flfEC(T) and f i s a bounded function}, Cc (T) = {fI fEC(T) and f has a compact support}. If fEC(T) then Z(f)--{tETlf(t)=O}, Zc(f)=TN.Z(f) . X(T) is a vector lattice iff X(T)cC(T), f, gEX(T) and LER imply fVgEX(T), fAgEX(T), f +gEX(T), ~fEX(T). Let 1"1, T~ be Tihonov spaces, X(T1), Y(T2) two vector lattices, cp: X(TO~ -~ Y(T.,) is said to be a linear lattice homomorphism iff ~o is a lattice homomorphism and xl,x~EX(T) implies ~o(x~-x~)=cp(xO-cp(x~). X(T) is said to be a completely regular vector lattice over Tiff, whenever F is a closed set and t is a point in its complement, there exists a function fEX(T) such that 0~f_-<l, f (F)={0} and f(Ut)={l}, where Ut is a neighbourhood of t. We shall omit the easy proof of the following properties: 1. Cc(X), C*(X), C(X) are vector lattices, Cc(X)cC*(X)cC(X). 2. Let cp: X(T)-~R be a linear lattice homomorphism, i.e. let cp be a lattice homomorphism and q~(xl-x2)--cp(xO-~o(x~) for x~, x2EX(T), a) If f , gEKer cp, then f g E K e r cp. b) If fEKer cp, gEX(T), ]g[<=nlfl, then gEKer ~0.