Tensor Products of Convex Cones

TL;DR

This paper develops a comprehensive theory of tensor products of convex cones, extending previous results to infinite-dimensional and non-Archimedean cones, and introduces new formulas and properties for these tensor products.

Contribution

It generalizes tensor product theory of convex cones to all cones and spaces, providing new formulas, properties, and confirming a conjecture for broad classes of cones.

Findings

Tensor product of two symmetric convex sets preserves proper faces.

Projective/injective cones have analogous mapping properties to norms.

The projective cone is closed in finite-dimensional spaces and mostly strictly contained in the injective cone.

Abstract

Tensor products of convex cones have recently come up in different areas, ranging from functional analysis and operator theory to approximation theory and theoretical physics. However, most of the existing literature focuses either on Archimedean lattice cones or on closed proper cones in finite-dimensional spaces, thereby excluding many cones, including even standard cones such as an infinite-dimensional positive semidefinite cone. For general cones, results are few and far between, and many basic questions remain unanswered. In this memoir, we develop the theory of tensor products of convex cones in full generality, with no restrictions on the cones or the ambient spaces. We generalize a few known results to the general case, and we prove many results which are altogether new. Our main contributions are: (i) 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; (iii) We prove that the projective/injective tensor product of two closed proper cones is contained in a closed proper cone; (iv) We show how to construct faces of the projective/injective cone from faces of the base cones. As an application, we also show that the tensor product of two symmetric convex sets preserves proper faces; (v) For closed cones in finite-dimensional spaces, we show that the projective cone is closed, and almost always strictly contained in the injective cone, thereby confirming a conjecture of Barker for nearly all convex cones. As this manuscript was being written, this last result was superseded by simultaneous discovery by Aubrun, Lami, Palazuelos and Pl\'avala, who independently proved Barker's conjecture in full generality. We recover their result for a large class of cones, using completely different techniques.