Operators on Banach Lattices as Weighted Compositions

Abstract

In this paper we study lattice (Riesz) homomorphisms or, more generally, disjointness preserving (Lamperti, see [2]) operators between real Banach lattices having locally compact representation spaces in the sense of Schaefer [12, III 5.4]. We recall that a Banach lattice E is said to have a locally compact representation space if E contains a copy of C k (X), the continuous real-valued functions on a locally compact space X with compact support, as a dense (order) ideal. This broad class of Banach lattices includes, for example, all Banach lattices with quasi-interior points, topological orthogonal systems or topological order partitions. Such lattices have been studied in [3,4,11,12].These Banach lattices can be represented as continuous extended real-valued functions on their locally compact representation spaces (for example see [3,4]). Theorems 1 and 2 below describe the lattice homomorphisms or disjointness preserving operators on these Banach lattices as weighted composition operators with respect to the representation spaces, generalizing results known for the function lattices of the type C(K) where K is compact (see [15]). and for quasi-interior point spaces (see [10]). These results should also be compared with those of Abramovich [1]. We show that, if T is a lattice homomorphism between Banach lattices E and F, then, given / in E, Tf is of the form r(/o </>), where r is a continuous real-valued function on the representation space for F and ^ is an essentially continuous map between the appropriate representation spaces. These results permit an analysis of the principal order ideal ZT generated by a lattice homomorphism T. Corollary 1 states that 3T is equal to the orthomorphisms composed with T. It also generalizes a result of Nagel [10, 3.3], and complements Haid's Theorem 2.7 in [6]. Theorem 5 states that each positive element in the closure of ST can be expressed as the composition with T of what we call a' partial orthomorphism'. This result and Corollary 3 are analogues of a Radon-Nikodym type theorem proved by Luxemburg and Schep (see [8,9]). Corollary 4 provides sufficient conditions for the principal ideal &~ generated by T to be closed. PreliminariesGiven a Banach lattice E with a locally compact representation space X, we shall identify E with a space of continuous extended real-valued functions on X (each finite on a dense set) containing C k (X) as a dense order ideal [3]. We recall [3] that E has a locally compact representation space if and only if E contains a topological order partition (t.o.p.). A special case of a t.o.p. is a quasi-interior point, a positive element generating a dense order ideal.For a continuous function r ^ 0 on X, we let P r = {xeX: r(x) > 0}, Z r = {xeX: r(x) = 0} and we denote the interior of Z r by Z°r. We shall use the fact that P r U Z°r is dense in X.

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