Topological concepts in partially ordered vector spaces

In the context of partially ordered vector spaces one encounters different sorts of order convergence and order topologies. This article investigates these notions and their relations. In particular, we study and relate the order topology presented by Floyd, Vulikh and Dobbertin, the order bound topology studied by Namioka and the concept of order convergence given in the works of Abramovich, Sirotkin, Wolk and Vulikh. We prove that the considered topologies disagree for all infinite dimensional Archimedean vector lattices that contain order units. For reflexive Banach spaces equipped with ice cream cones we show that the order topology, the order bound topology and the norm topology agree and that order convergence is equivalent to norm convergence.

Operators in pre-Riesz spaces: moduli and homomorphisms

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.

Order continuity from a topological perspective

We study three types of order convergence and related concepts of order continuous maps in partially ordered sets, partially ordered abelian groups, and partially ordered vector spaces, respectively. An order topology is introduced such that in the latter two settings under mild conditions order continuity is a topological property. We present a generalisation of the Ogasawara theorem on the structure of the set of order continuous operators.

On the Linearity of Order-isomorphisms

A basic problem in the theory of partially ordered vector spaces is to characterise those cones on which every order-isomorphism is linear. We show that this is the case for every Archimedean cone that equals the inf-sup hull of the sum of its engaged extreme rays. This condition is milder than existing ones and is satisfied by, for example, the cone of positive operators in the space of bounded self-adjoint operators on a Hilbert space. We also give a general form of order-isomorphisms on the inf-sup hull of the sum of all extreme rays of the cone, which extends results of Artstein–Avidan and Slomka to infinite-dimensional partially ordered vector spaces, and prove the linearity of homogeneous order-isomorphisms in a variety of new settings.