Functional representation of vector lattices II

Abstract

Schaefer develops a representation for Banach lattices with a "quasi-interior positive element" as continuous extended real-valued functions on a compact space, finite except on rare subsets [2, III 4.5]. This may be obtained by specialization from the following purely algebraic: An Archimedean vector lattice with weak order unit is represented in this way on the compact space of its maximal ideals. Indeed, a normed vector lattice is Archimedean [2, p. 81] and a quasi-interior element is a weak order unit [2, p. 97]. Moreover, every Banach lattice is relatively uniformly complete [2, II 7.2 Corollary] and with this additional axiom the ideal generated by the unit will induce all bounded continuous functions as in [2, III 4.5]. Finally, this ideal is always order dense and Schaefer's requirement that the unit be quasi-interior is just what makes it topologically (i.e. norm) dense. By first embedding the vector lattice in its Dedekind-MacNeil le completion this construction also furnishes a new proof of the classical representation of such vector lattices as continuous functions on extremaly disconnected compact spaces [1, 50.9], [3, V4.1]. A lattice-ordered (abbreviated l-) group is a group (here assumed commutative and written additively) equipped with a lattice order, such that the group translations by each element are lattice automorphisms. One introduces the positive - g + := g v 0 and negative - g-:= (-g)v 0 - parts of each element g and verifies that their difference is g - g + g -= g + (-g v 0)= 0 v g = g +, and that they are disjoint 0 = -g- + g- = (g/x 0) + g- = g + A g-. It follows that homomorphisms are recoverable from their restrictions to the positive cones {g : g > 0}, which are additive 0-preserving lattice morphisms; and every such t) extends uniquely to a homomorphism between the/-groups of their differences (indeed, even to maximally defined morphisms to certain images which, like the extended reals, are not additively closed). These morphisms have kernels which are "ideals" - i.e. additively closed subsets closed for meet with every element in the cone - which determine the quotient surjection since f is identified with f/x g just when f - f/x g is sent on 0. Every ideal is a kernel, since the elements in the /-group between some ideal element and its negative form a solid subgroup which meets 1) It suffices to have it a semilattice monoid morphism since e.g. f+ g-(g A f)= f + g + (-g v - f) = f v g expresses one of v, /x in terms of the other; and even just disjointnesspreserving and subtractive since then 0 = q) (f - f/x g) A (p (g - f/x g) = [(P (f) - q) (f/x g)] A [r (g) -- ~o (f A g)] = [~o (f) A ~ (g)] - q~ (f ^ g).