Geometric duality theory of cones in dual pairs of vector spaces

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.