Operators in pre-Riesz spaces: moduli and homomorphisms

Positivity, vol. 25, pp. 2099–2136

Abstract

Abstract We focus on two topics that are related to moduli of elements in partially ordered vector spaces. First, we relate operators that preserve moduli to generalized notions of lattice homomorphisms, such as Riesz homomorphisms, Riesz* homomorphisms, and positive disjointness preserving operators. We also consider complete Riesz homomorphisms, which generalize order continuous lattice homomorphisms. Second, we characterize elements with a modulus by means of disjoint elements and apply this result to obtain moduli of functionals and operators in various settings. On spaces of continuous functions, we identify those differences of Riesz* homomorphisms that have a modulus. Many of our results for pre-Riesz spaces of continuous functions lead to results on order unit spaces, where the functional representation is used.

Authors 3

1. Anke Kalauch

   Technische Universität Dresden

   Affiliation as printed

   Fakultät Mathematik, Institut für Analysis, TU Dresden, 01062, Dresden, Germany

2. Janko Stennder corresponding

   Technische Universität Dresden

   Affiliation as printed

   Fakultät Mathematik, Institut für Analysis, TU Dresden, 01062, Dresden, Germany

3. Onno van Gaans Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University, P.O. Box 9512, 2300, RA, Leiden, The Netherlands