Operators on anti-dual pairs: Generalized Krein--von Neumann extension

TL;DR

This paper extends the classical theory of positive operators to anti-dual pairs, providing a comprehensive extension framework applicable in spaces lacking Hilbert space structure or normability, with applications in noncommutative integration.

Contribution

It introduces a generalized extension theory for positive operators on anti-dual pairs, overcoming limitations of Hilbert space frameworks and enabling applications in broader mathematical contexts.

Findings

Characterization of operators admitting positive extensions

Analysis of properties like commutation and minimality of extensions

Application to noncommutative integration theory

Abstract

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.