Positivity 7: 141–148, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
On Ideal Operators
Department of Mathematics, Faculty of Arts and Sciences, Gazi University, Teknikokullar 06500
Ankara, Turkey. E-mail:email@example.com
Abstract. Let E, F be Archimedean Riesz spaces. We consider operators that map ideals of E to
ideals of F and operators T for which, T
(I ) is an ideal in E, for each ideal I in F . We study the
properties of such operators and investigate their relation to disjointness preserving operators.
For Archimedean Riesz spaces E,F, L
(E, F ) will denote the space of order
bounded operators from E into F . The collection of all operators satisfying π(x) ⊥
y whenever x ⊥ y will be denoted by Orth (E). Orth (E) is an Archimedean
Riesz space. The ideal generated by the identity operator I in Orth (E) is called
the ideal centre of E and will be denoted by Z(E).BothofZ(E) and Orth (E)
are f -algebras. An operator T : E → F satisfying Tx ⊥ Ty whenever x ⊥
y is called a disjointness preserving operator. A positive operator T , satisfying
T [0,x]=[0,Tx] for each x ∈ E
is called an interval preserving (Maharam)
operator. We refer to ,  and  for deﬁnitions and notation not explained
2. Ideal Operators
DEFINITION 2.1. An operator T between Riesz spaces E, F is called inverse
ideal operator if T
(J ) is an ideal in E whenever J is an ideal in F .
DEFINITION 2.2. An operator T between Riesz spaces E, F is called an ideal
operator if T(I)is an ideal in F for each ideal I in E.
PROPOSITION 2.3. Let E, F be Riesz spaces and T : E → F be a linear
operator and I
be order ideal generated by x.Then
(i) A necessary and sufﬁcient condition for T to be an ideal operator is that I
) for each x ∈ E.
(ii) A necessary and sufﬁcient condition for T to be an inverse ideal operator is
) ⊆ I
for each x ∈ E.
Proof. (i) Let T be an ideal operator. For each x ∈ E, T(I
) is an ideal in F
containing Tx. Hence I