Tensor Products of Convex Cones, Part I: Mapping Properties, Faces, and Semisimplicity

TL;DR

This paper explores the properties of tensor products of convex cones in ordered vector spaces, extending the theory from algebraic to locally convex tensor products and analyzing their structure and mapping behaviors.

Contribution

It introduces new parallels between tensor product cones and normed space theory, providing formulas, characterizations, and construction methods for faces and extremal rays.

Findings

Mapping properties analogous to normed spaces

Formulas for lineality space and properness conditions

Complete characterization of extremal rays

Abstract

The tensor product of two ordered vector spaces can be ordered in more than one way, just as the tensor product of normed spaces can be normed in multiple ways. Two natural orderings have received considerable attention in the past, namely the ones given by the projective and injective (or biprojective) cones. This paper aims to show that these two cones behave similarly to their normed counterparts, and furthermore extends the study of these two cones from the algebraic tensor product to completed locally convex tensor products. The main results in this paper are the following: (i) drawing parallels with the normed theory, we show that the projective/injective cone has mapping properties analogous to those of the projective/injective norm; (ii) we establish direct formulas for the lineality space of the projective/injective cone, in particular providing necessary and sufficient conditions for the cone to be proper; (iii) we show how to construct faces of the projective/injective cone from faces of the base cones, in particular providing a complete characterization of the extremal rays of the projective cone; (iv) we prove that the projective/injective tensor product of two closed proper cones is contained in a closed proper cone (at least in the algebraic tensor product).