Classes of operators preserved by extensions or liftings

A standard way to obtain extensions (resp. liftings) of operators is by making the so-called operations of push-out (resp. pull-back). In this paper we study the preservation of some classes of operators associated with an operator ideal A under push-out extensions or pull-back liftings. We show several examples of classical operator ideals whose associated classes are preserved, we prove that the preservation of those classes under push-out extension or pull-back lifting implies that the space ideal of A satisfies the 3-space property, and we derive some results for A that are useful in the study of commutative diagrams of operators.