  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.
