Dilations, inclusions of matrix convex sets, and completely positive maps

TL;DR

This paper explores the geometry of matrix convex sets, their relation to completely positive maps, and establishes dilation and inclusion results with explicit constants, advancing understanding in operator theory and spectrahedral problems.

Contribution

It introduces geometric conditions for matrix convex set inclusions, constructs a self-dual matrix convex set, and links these to dilation theory and spectrahedral inclusion problems.

Findings

Constants depend on the number of variables, not ranks.

Existence of a unique self-dual matrix convex set, the matrix ball.

Inclusion results with sharp constants under symmetry conditions.

Abstract

A matrix convex set is a set of the form $\mathcal{S} = \cup_{n\geq 1}\mathcal{S}_n$ (where each $\mathcal{S}_n$ is a set of $d$-tuples of $n \times n$ matrices) that is invariant under UCP maps from $M_n$ to $M_k$ and under formation of direct sums. We study the geometry of matrix convex sets and their relationship to completely positive maps and dilation theory. Key ingredients in our approach are polar duality in the sense of Effros and Winkler, matrix ranges in the sense of Arveson, and concrete constructions of scaled commuting normal dilation for tuples of self-adjoint operators, in the sense of Helton, Klep, McCullough and Schweighofer. Given two matrix convex sets $\mathcal{S} = \cup_{n \geq 1} \mathcal{S}_n,$ and $\mathcal{T} = \cup_{n \geq 1} \mathcal{T}_n$, we find geometric conditions on $\mathcal{S}$ or on $\mathcal{T}$, such that $\mathcal{S}_1 \subseteq \mathcal{T}_1$ implies that $\mathcal{S} \subseteq C\mathcal{S}$ for some constant $C$. For instance, under various symmetry conditions on $\mathcal{S}$, we can show that $C$ above can be chosen to equal $d$, the number of variables, and in some cases this is sharp. We also find an essentially unique self-dual matrix convex set $\mathcal{D}$, the self-dual matrix ball, for which corresponding inclusion and dilation results hold with constant $C=\sqrt{d}$. Our results have immediate implications to spectrahedral inclusion problems studied recently by Helton, Klep, McCullough and Schweighofer. Our constants do not depend on the ranks of the pencils determining the free spectrahedra in question, but rather on the "number of variables" $d$. There are also implications to the problem of existence of (unital) completely positive maps with prescribed values on a set of operators.