# A relationship between the space of orthomorphisms and the centre of a vector lattice

A relationship between the space of orthomorphisms and the centre of a vector lattice When are so happy in a vector lattice that all band preserving linear operators turn out to be in the ideal centre? This question was raised by Wickstead (Representation and duality in Representation and duality Riesz spaces. Compos Math 35(3):225–238, 1977) (though not explicitly stated) and by the first author and Chil and Meyer (On the centre of a vector lattice. Indag Math 23:167–183, 2012). The answer depends on the vector lattice in which the operator in question acts. However, in this article we focus our attention on this question. First, we give a complete description of those vector lattices \$\$E\$\$ E with the property that every orthomorphism on \$\$E\$\$ E is a central operator. Secondly, we provide a counterexample to the main result, about Wickstead’s question, in a recent paper of Toumi [see Theorem 3, When orthomorphisms are in the ideal center, in Positivity (2014, in press). (Published online: 11 Dec 2013)]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# A relationship between the space of orthomorphisms and the centre of a vector lattice

, Volume 19 (3) – Sep 16, 2014
9 pages

/lp/springer_journal/a-relationship-between-the-space-of-orthomorphisms-and-the-centre-of-a-orXh0yqV2j
Publisher
Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-014-0308-2
Publisher site
See Article on Publisher Site

### Abstract

When are so happy in a vector lattice that all band preserving linear operators turn out to be in the ideal centre? This question was raised by Wickstead (Representation and duality in Representation and duality Riesz spaces. Compos Math 35(3):225–238, 1977) (though not explicitly stated) and by the first author and Chil and Meyer (On the centre of a vector lattice. Indag Math 23:167–183, 2012). The answer depends on the vector lattice in which the operator in question acts. However, in this article we focus our attention on this question. First, we give a complete description of those vector lattices \$\$E\$\$ E with the property that every orthomorphism on \$\$E\$\$ E is a central operator. Secondly, we provide a counterexample to the main result, about Wickstead’s question, in a recent paper of Toumi [see Theorem 3, When orthomorphisms are in the ideal center, in Positivity (2014, in press). (Published online: 11 Dec 2013)].

### Journal

PositivitySpringer Journals

Published: Sep 16, 2014

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just \$49/month

### Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

### Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

### Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

### Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

DeepDyve

DeepDyve

### Pro

Price

FREE

\$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations