  The main aim of this paper is to generalize the classical concept of positive
operator, and to develop a general extension theory, which overcomes not only
the lack of a Hilbert space structure, but also the lack of a normable
topology. The concept of anti-duality carries an adequate structure to define
positivity in a natural way, and is still general enough to cover numerous
important areas where the Hilbert space theory cannot be applied. Our running
example -- illustrating the applicability of the general setting to spaces
bearing poor geometrical features -- comes from noncommutative integration
theory. Namely, representable extension of linear functionals of involutive
algebras will be governed by their induced operators. The main theorem, to
which the vast majority of the results is built, gives a complete and
constructive characterization of those operators that admit a continuous
positive extension to the whole space. Various properties such as commutation,
or minimality and maximality of special extensions will be studied in detail.
