Extensions of positive operators and functionals

TL;DR

This paper studies positive linear operators and functionals on Banach spaces, providing constructive characterizations for their bounded extensions, compactness, and conditions for representability.

Contribution

It offers a new constructive characterization of positive operator extensions and applies it to positive functionals on Banach *-algebras.

Findings

Characterization of bounded positive extensions

Conditions for compactness of extended operators

Criteria for representable positive extensions

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 goal of this paper is to investigate of this kind of operators. The main theorem is a constructive characterization of the bounded positive extendibility of these linear mappings. From this result we can characterize the compactness of the extended operators and that when the positive extensions have closed ranges. As a main application of our general extension theorem, we present some necessary and sufficient conditions that a positive functional defined on a left ideal of a Banach $^*$-algebra admits a representable positive extension. The approach we use here is completely constructive.