Riesz spaces of order bounded disjointness preserving operators

Amor, Fethi Ben

Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 4, pp. 607-622.

Résumé

Let $L$, $M$ be Archimedean Riesz spaces and $\Cal L_{b}(L,M)$ be the ordered vector space of all order bounded operators from $L$ into $M$. We define a Lamperti Riesz subspace of $\Cal L_{b}(L,M)$ to be an ordered vector subspace $\Cal L$ of $\Cal L_{b}(L,M)$ such that the elements of $\Cal L$ preserve disjointness and any pair of operators in $\Cal L$ has a supremum in $\Cal L_{b}(L,M)$ that belongs to $\Cal L$. It turns out that the lattice operations in any Lamperti Riesz subspace $\Cal L$ of $\Cal L_{b}(L,M)$ are given pointwise, which leads to a generalization of the classic Radon-Nikod'ym theorem for Riesz homomorphisms. We then introduce the notion of maximal Lamperti Riesz subspace of $\Cal L_{b}(L,M)$ as a generalization of orthomorphisms. In this regard, we show that any maximal Lamperti Riesz subspace of $\Cal L_{b}(L,M)$ is a band of $\Cal L_{b}(L,M)$, provided $M$ is Dedekind complete. Also, we extend standard transferability theorems for orthomorphisms to maximal Lamperti Riesz subspace of $\Cal L_{b}(L,M)$. Moreover, we give a complete description of maximal Lamperti Riesz subspaces on some continuous function spaces.

Détail

Classification : 06F20, 46A32, 46A40, 47B65
Keywords: continuous functions spaces; disjointness preserving operator; Lamperti Riesz subspace; order bounded operator; orthomorphism; Radon-Nikod'ym; Riesz space