| Title:
|
Riesz spaces of order bounded disjointness preserving operators (English) |
| Author:
|
Amor, Fethi Ben |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
48 |
| Issue:
|
4 |
| Year:
|
2007 |
| Pages:
|
607-622 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $L$, $M$ be Archimedean Riesz spaces and $\Cal L_{b}(L,M)$ be the ordered vector space of all order bounded operators from $L$ into $M$. We define a Lamperti Riesz subspace of $\Cal L_{b}(L,M)$ to be an ordered vector subspace $\Cal L$ of $\Cal L_{b}(L,M)$ such that the elements of $\Cal L$ preserve disjointness and any pair of operators in $\Cal L$ has a supremum in $\Cal L_{b}(L,M)$ that belongs to $\Cal L$. It turns out that the lattice operations in any Lamperti Riesz subspace $\Cal L$ of $\Cal L_{b}(L,M)$ 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 $\Cal L_{b}(L,M)$ as a generalization of orthomorphisms. In this regard, we show that any maximal Lamperti Riesz subspace of $\Cal L_{b}(L,M)$ is a band of $\Cal L_{b}(L,M)$, provided $M$ is Dedekind complete. Also, we extend standard transferability theorems for orthomorphisms to maximal Lamperti Riesz subspace of $\Cal L_{b}(L,M)$. Moreover, we give a complete description of maximal Lamperti Riesz subspaces on some continuous function spaces. (English) |
| Keyword:
|
continuous functions spaces |
| Keyword:
|
disjointness preserving operator |
| Keyword:
|
Lamperti Riesz subspace |
| Keyword:
|
order bounded operator |
| Keyword:
|
orthomorphism |
| Keyword:
|
Radon-Nikod'ym |
| Keyword:
|
Riesz space |
| MSC:
|
06F20 |
| MSC:
|
46A32 |
| MSC:
|
46A40 |
| MSC:
|
47B65 |
| idZBL:
|
Zbl 1199.06071 |
| idMR:
|
MR2375162 |
| . |
| Date available:
|
2009-05-05T17:05:05Z |
| Last updated:
|
2012-05-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119684 |
| . |
| Reference:
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| . |
