Dilations, inclusions of matrix convex sets, and completely positive maps
Kenneth R. Davidson, Adam Dor-On, Orr Shalit, Baruch Solel
TL;DR
The paper develops a unifying operator-theoretic framework for matrix convex sets, polar duality, matrix ranges, and dilations, connecting interpolation of $d$-tuples by UCP maps to containment relations between matrix convex sets. It proves that a unital CP interpolation exists precisely when the matrix range of the target tuple lies inside that of the source, and provides finite-dimensional, symmetry-enhanced, and spectral-dilative methods to obtain scaled inclusions like $\,\mathcal{S}\subseteq d\mathcal{T}$ or $\,\mathcal{S}\subseteq\sqrt{d}\mathcal{D}$ under various symmetry assumptions. The work gives sharp constants (notably $C=d$ and $C=\sqrt{d}$ in self-dual cases) and constructs explicit dilations to commuting normal tuples to certify inclusions, with wide implications for spectrahedral containment problems and completely positive maps. It also introduces a self-dual matrix ball and analyzes matrix balls, diamonds, and tight-frame-generated polytopes, demonstrating both the reach and limits of matricial relaxation techniques in noncommutative optimization and operator-system theory.
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.