Geometric duality theory of cones in dual pairs of vector spaces

TL;DR

This paper extends the geometric duality theory of cones from real pre-ordered Banach spaces to more general Banach spaces with multiple cones, establishing duality relations for geometric properties and applying them to C*-algebras.

Contribution

It introduces a generalized framework for geometric duality of cones in Banach spaces with multiple cones, beyond classical single-cone settings.

Findings

Duality between normality and conormality in dual pairs of vector spaces.

Generalization of classical duality results to spaces with multiple cones.

Application to geometric properties of cones in C*-algebras.

Abstract

This paper will generalize what may be termed the "geometric duality theory" of real pre-ordered Banach spaces which relates geometric properties of a closed cone in a real Banach space, to geometric properties of the dual cone in the dual Banach space. We show that geometric duality theory is not restricted to real pre-ordered Banach spaces, as is done classically, but can be extended to real Banach spaces endowed with arbitrary collections of closed cones. We define geometric notions of normality, conormality, additivity and coadditivity for members of dual pairs of real vector spaces as certain possible interactions between two cones and two convex convex sets containing zero. We show that, thus defined, these notions are dual to each other under certain conditions, i.e., for a dual pair of real vector spaces $(Y,Z)$, the space $Y$ is normal (additive) if and only if its dual $Z$ is conormal (coadditive) and vice versa. These results are set up in a manner so as to provide a framework to prove results in the geometric duality theory of cones in real Banach spaces. As an example of using this framework, we generalize classical duality results for real Banach spaces pre-ordered by a single closed cone, to real Banach spaces endowed with an arbitrary collections of closed cones. As an application, we analyze some of the geometric properties of naturally occurring cones in C*-algebras and their duals.