Riesz spaces of order bounded disjointness preserving operators

Fethi Ben Amor

Commentationes Mathematicae Universitatis Carolinae (2007)

  • Volume: 48, Issue: 4, page 607-622
  • ISSN: 0010-2628

Abstract

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Let L , M be Archimedean Riesz spaces and 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 ℒ b ( L , M ) to be an ordered vector subspace ℒ of ℒ b ( L , M ) such that the elements of ℒ preserve disjointness and any pair of operators in ℒ has a supremum in ℒ b ( L , M ) that belongs to ℒ . It turns out that the lattice operations in any Lamperti Riesz subspace ℒ of 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 ℒ b ( L , M ) as a generalization of orthomorphisms. In this regard, we show that any maximal Lamperti Riesz subspace of ℒ b ( L , M ) is a band of ℒ b ( L , M ) , provided M is Dedekind complete. Also, we extend standard transferability theorems for orthomorphisms to maximal Lamperti Riesz subspace of ℒ b ( L , M ) . Moreover, we give a complete description of maximal Lamperti Riesz subspaces on some continuous function spaces.

How to cite

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Amor, Fethi Ben. "Riesz spaces of order bounded disjointness preserving operators." Commentationes Mathematicae Universitatis Carolinae 48.4 (2007): 607-622. <http://eudml.org/doc/250200>.

@article{Amor2007,
abstract = {Let $L$, $M$ be Archimedean Riesz spaces and $\mathcal \{L\}_\{b\}(L,M)$ be the ordered vector space of all order bounded operators from $L$ into $M$. We define a Lamperti Riesz subspace of $\mathcal \{L\}_\{b\}(L,M)$ to be an ordered vector subspace $\mathcal \{L\}$ of $\mathcal \{L\}_\{b\}(L,M)$ such that the elements of $\mathcal \{L\}$ preserve disjointness and any pair of operators in $\mathcal \{L\}$ has a supremum in $\mathcal \{L\}_\{b\}(L,M)$ that belongs to $\mathcal \{L\}$. It turns out that the lattice operations in any Lamperti Riesz subspace $\mathcal \{L\}$ of $\mathcal \{L\}_\{b\}(L,M)$ 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 $\mathcal \{L\}_\{b\}(L,M)$ as a generalization of orthomorphisms. In this regard, we show that any maximal Lamperti Riesz subspace of $\mathcal \{L\}_\{b\}(L,M)$ is a band of $\mathcal \{L\}_\{b\}(L,M)$, provided $M$ is Dedekind complete. Also, we extend standard transferability theorems for orthomorphisms to maximal Lamperti Riesz subspace of $\mathcal \{L\}_\{b\}(L,M)$. Moreover, we give a complete description of maximal Lamperti Riesz subspaces on some continuous function spaces.},
author = {Amor, Fethi Ben},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {continuous functions spaces; disjointness preserving operator; Lamperti Riesz subspace; order bounded operator; orthomorphism; Radon-Nikod'ym; Riesz space; spaces of continuous functions; disjointness-preserving operator; Lamperti-Riesz subspace},
language = {eng},
number = {4},
pages = {607-622},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Riesz spaces of order bounded disjointness preserving operators},
url = {http://eudml.org/doc/250200},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Amor, Fethi Ben
TI - Riesz spaces of order bounded disjointness preserving operators
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 4
SP - 607
EP - 622
AB - Let $L$, $M$ be Archimedean Riesz spaces and $\mathcal {L}_{b}(L,M)$ be the ordered vector space of all order bounded operators from $L$ into $M$. We define a Lamperti Riesz subspace of $\mathcal {L}_{b}(L,M)$ to be an ordered vector subspace $\mathcal {L}$ of $\mathcal {L}_{b}(L,M)$ such that the elements of $\mathcal {L}$ preserve disjointness and any pair of operators in $\mathcal {L}$ has a supremum in $\mathcal {L}_{b}(L,M)$ that belongs to $\mathcal {L}$. It turns out that the lattice operations in any Lamperti Riesz subspace $\mathcal {L}$ of $\mathcal {L}_{b}(L,M)$ 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 $\mathcal {L}_{b}(L,M)$ as a generalization of orthomorphisms. In this regard, we show that any maximal Lamperti Riesz subspace of $\mathcal {L}_{b}(L,M)$ is a band of $\mathcal {L}_{b}(L,M)$, provided $M$ is Dedekind complete. Also, we extend standard transferability theorems for orthomorphisms to maximal Lamperti Riesz subspace of $\mathcal {L}_{b}(L,M)$. Moreover, we give a complete description of maximal Lamperti Riesz subspaces on some continuous function spaces.
LA - eng
KW - continuous functions spaces; disjointness preserving operator; Lamperti Riesz subspace; order bounded operator; orthomorphism; Radon-Nikod'ym; Riesz space; spaces of continuous functions; disjointness-preserving operator; Lamperti-Riesz subspace
UR - http://eudml.org/doc/250200
ER -

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