Positivity 7: 141–148, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands. 141 On Ideal Operators BAHRI TURAN Department of Mathematics, Faculty of Arts and Sciences, Gazi University, Teknikokullar 06500 Ankara, Turkey. E-mail:bturan@gazi.edu.tr 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 −1 (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. 1. Introduction For Archimedean Riesz spaces E,F , L b (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). Both of Z(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 [2], [8] and [10] for definitions and notation not explained here. 2. Ideal Operators DEFINITION 2.1. An operator T between Riesz spaces E,F is called inverse ideal operator if T −1 (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 x be order ideal generated by x . Then (i) A necessary and sufficient condition for T to be an ideal operator is that I Tx ⊆ T (I x ) for each x ∈ E. (ii) A necessary and sufficient condition for T to be an inverse ideal operator is that T (I x ) ⊆ I Tx for each x ∈ E. Proof. (i) Let T be an ideal operator. For each x ∈ E, T (I x ) is an ideal in F containing Tx . Hence I Tx ⊆ T (I x ).