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.

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.

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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