Positivity 7: 125–133, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
On Ideals Generated by Positive Operators
and A. UYAR
Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey.
Department of Math Education, Gazi University, Ankara, Turkey. (e-mail: email@example.com)
Abstract. Algebra structure of principle ideals of order bounded operators is studied.
The Riesz spaces in this paper are assumed to have separating order duals. Let E
be an ordered vector space, L
(E) be the space of order bounded operators. The
order dual of E will be denoted by E
. Z(E) will denote the ideal centre, Orth
E will denote the space of all orthomorphisms of E. In all undeﬁned terminology
concerning Riesz spaces we will adhere to .
DEFINITION 1.1. A Riesz space E, is said to have topologically full centre if,
for each pair x, y in E with 0 y x, there exists a net (π
) in Z(E) with
I for each α, such that π
x → y in σ(E,E
Banach lattices with topologically full centre were initiated in . These spaces
were also studied in ,  and . The class of Riesz spaces and the class of
Banach lattices that have topologically full centre are quite large. σ -Dedekind com-
plete Riesz spaces have topologically full centres. However, not all Riesz spaces
have topologically full centres.
A positively generated subspace F of an ordered vector space E is called an
ideal if x ∈ F
and 0 y x imply y ∈ F .IfT : E → E is an order bounded
operator, we denote the ideal generated by T in L
(E) by A
the band generated by T .
2. Main results
The property that A
is an algebra is independent from T being in Z(E) or in
EXAMPLE 2.1. The operator S : l
deﬁned by S(x
) = (
, 0, ··· ,
0 ···) exhibits a positive operator S where A
has a non-trivial algebra structure.
On the other hand, S is not an orthomorphism as Se