Regular operators from and into a small Riesz space
type: Item
instance of: scholarly article
title: Regular operators from and into a small Riesz space
zbMATH Open document ID: 0753.46009
DOI: 10.1016/0019-3577(91)90014-X
published in: Indagationes Mathematicae. New Series
publication date: 1992-09-27T00:00:00Z
review text: The authors consider in great detail the spaces \(L^ r\) (resp. \(L^ b\)) of all regular (resp. order bounded) operators from and into the smallest possible non-Dedekind complete Riesz space \(\ell_ 0^ \infty\) of all eventually constant sequences. We confine ourselves to mentioning some of the most striking results: (i) if \(E\) is a uniformly complete Riesz space, then \(L^ r(E,\ell_ 0^ \infty)=L^ b(E,\ell_ 0^ \infty)\); furthermore, \(L^ r(E,\ell_ 0^ \infty)\) is a Riesz space in which all lattice operations may be computed with the Riesz-Kantorovich formula. (ii) \(L^ r(\ell_ 0^ \infty,F)=L^ b(\ell_ 0^ \infty,F)\) for all Archimedean Riesz spaces \(F\); moreover, \(L^ r(\ell_ 0^ \infty,F)\) is a Riesz space if and only if \(F\) is Dedekind \(\sigma\)- complete. Hence, \(L^ r(\ell_ 0^ \infty)\) is not a Riesz space. It even fails to have the Riesz decomposition property. For the many other results we refer to this interesting paper itself.
reviewed by: Charles B. Huijsmans
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 47B65
zbMATH DE Number: 57308
zbMATH Keywords: regular operator
zbMATH Keywords: small Riesz space
zbMATH Keywords: order bounded operators
zbMATH Keywords: Dedekind complete Riesz space
zbMATH Keywords: Riesz-Kantorovich formula
zbMATH Keywords: Dedekind \(\sigma\)- complete
zbMATH Keywords: Riesz decomposition property
author: Y. A. Abramovich
author: A. W. Wickstead
MaRDI profile type: Publication
cites work: Q3683058
cites work: On a question of Fremlin concerning order bounded and regular operators
cites work: Small Riesz spaces
cites work: An Example in the Space of Bounded Operators from C(X) to C(Y)
cites work: Q5638775
cites work: Q3738029
cites work: THE REGULARITY OF ORDER BOUNDED OPERATORS INTO C(K)
cites work: Q3669960
full work available at URL: 0019-3577(91)90014-x
OpenAlex ID: W2070410302
description: scientific article; zbMATH DE number 57308