scholarly journals Topological concepts in partially ordered vector spaces

Positivity ◽  
2021 ◽  
Author(s):  
T. Hauser
Keyword(s):  
Order Topology ◽  
Vector Spaces ◽  
Order Convergence ◽  
Norm Topology ◽  
Reflexive Banach Spaces ◽  
Infinite Dimensional ◽  
Partially Ordered ◽  
Vector Lattices ◽  
Ordered Vector Spaces ◽  
Partially Ordered Vector Spaces

AbstractIn 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.

scholarly journals On the Linearity of Order-isomorphisms

2020 ◽  
pp. 1-18
Author(s):  
Bas Lemmens ◽  
Onno van Gaans ◽  
Hendrik van Imhoff
Keyword(s):  
Basic Problem ◽  
Vector Spaces ◽  
Order Isomorphism ◽  
Infinite Dimensional ◽  
Partially Ordered ◽  
Order Isomorphisms ◽  
Ordered Vector Spaces ◽  
Partially Ordered Vector Spaces ◽  
Adjoint Operators ◽  
Extreme Rays

Abstract 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.

scholarly journals On the Biadjoint of Riesz-like homomorphisms on partially ordered vector spaces

1997 ◽  
Vol 56 (3) ◽  
pp. 473-481
Author(s):  
Gerard Buskes ◽  
Jamie Summerville
Keyword(s):  
Vector Spaces ◽  
New Technique ◽  
Partially Ordered ◽  
Ordered Vector Spaces ◽  
Partially Ordered Vector Spaces ◽  
A New Technique

We generalise to partially ordered vector spaces, with a new technique, Arendt's approach to Kim's characterisation of Riesz homomorphisms.

scholarly journals Integral representation theorems in partially ordered vector spaces

1991 ◽  
Vol 51 (2) ◽  
pp. 187-215 ◽  
Author(s):  
Panaiotis K. Pavlakos
Keyword(s):  
Integral Representation ◽  
Quantum Probability ◽  
Jordan Algebras ◽  
Vector Spaces ◽  
Type I ◽  
Representation Theorems ◽  
Partially Ordered ◽  
Ordered Vector Spaces ◽  
Partially Ordered Vector Spaces ◽  
Set Function

AbstractDefining a Radon-type integration process we extend the Alexandroff, Fichtengolts-KantorovichHildebrandt and Riesz integral representation theorems in partially ordered vector spaces.We also identify some classes of operators with other classes of operator-valued set functions, the correspondence between operator and operator-valued set function being given by integration.All these established results can be immediately applied in C* -algebras (especially in W* -algebras and AW* -algebras of type I), in Jordan algebras, in partially ordered involutory (O*-)algebras, in semifields, in quantum probability theory, as well as in the operator Feynman-Kac formula.

Minimal decompositions in partially ordered normed vector spaces

1968 ◽  
Vol 64 (4) ◽  
pp. 989-1000 ◽  
Author(s):  
A. J. Ellis
Keyword(s):  
Positive Cone ◽  
Intersection Property ◽  
Vector Spaces ◽  
Partially Ordered ◽  
Base Section ◽  
Ordered Vector Spaces ◽  
Partially Ordered Vector Spaces ◽  
Minimal Decomposition ◽  
Normed Vector

In this paper we study partially ordered vector spaces X whose positive cone K possesses a base which defines a norm in X. A positive decomposition x = y − z of the element x is said to be minimal if ‖x‖ = ‖y‖ + ‖z‖. We proved in (6) that the property that every element of X has a unique minimal decomposition is equivalent to an intersection property for homothetic translates of the base. Section 2 of the present paper analyses this intersection property in much more detail and discusses possible generalizations of it.

scholarly journals Order continuity from a topological perspective

Positivity ◽  
2021 ◽  
Author(s):  
T. Hauser ◽  
A. Kalauch
Keyword(s):  
Vector Spaces ◽  
Ordered Sets ◽  
Order Continuity ◽  
Partially Ordered ◽  
Mild Conditions ◽  
Ordered Vector Spaces ◽  
Partially Ordered Vector Spaces ◽  
Continuous Maps ◽  
Continuous Operators ◽  
Order Continuous

AbstractWe 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.

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