Local operators on Banach lattices

Abstract

The purpose of this paper is to study local operators on Banach lattices. A motivation for this study is the fact that an ordinary textbook on Banach lattices will contain approximately 20 identities and inequalities that are useful for the study, while a textbook on Banach algebras essentially contains only 2 inevitable identities, namely the associative and the distributive laws of the multiplication. The distributive law asserts however that the multiplication in the algebra is bilinear and is therefore probably more useful than all the identities concerning the lattice operations put together. The purpose of this paper is to reduce in some sense the study of Banach lattices to a study of (singly generated) Banach modules over a C(X)-space. We do this by proving on one hand that Banach lattice is a module over a suitable C(X)-space, and on the other hand every singly generated Banach module over a C(X)-space can given the structure of a Banach lattice. As a consequence we can also prove that the Banach space of all of functions from B and from the dual space B' is an AL-space in the sense of Kakutani. By essentially the same methods we also prove that if a given Banach space B is both a Banach lattice, and a real Banach algebra with unit, and if the structures are related by the condition that the algebraic product of lattice disjoint elements is 0, then B is a C(X)-space. We shall conclude this introduction with two remarks concerning the nature of our results. First we wish to emphasize that for the most important reflexive concrete Banach lattices like e.g. LP(R) our theorems will tell nothing new. What they do say is that the pointwise products give all of L l(R) and the set of local operators is exactly L (R). Our theorems thus tell that the abstract case is essentially the same as the concrete case. Furthermore, Our theorems should all be expected since the crucial norm condition on a Banach lattice is that IYI~lxl ~ IIYII~llxll so the operation of multiplying by y/x should normdecreasing and this is what we prove.