Extensions of positive operators and functionals

Abstract

We consider linear operators defined on a subspace of a complex Banach space into its topological antidual acting positively in a natural sense. The main theorem is a constructive characterization of the bounded positive extendibility of these linear mappings. In this frame we characterize operators with a compact or a closed range extension. As a main application of our general extension theorem, we present some necessary and sufficient conditions for a positive functional defined on a left ideal of a Banach ⁎-algebra to admit a representable positive extension.