Dilations, Inclusions of Matrix Convex Sets, and Completely Positive Maps

A matrix convex set is a set of the form S = U_n≥1S_n (where each S_n is a set of d-tuples of n×n matrices) that is invariant under unital completely positive 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 S = U_n≥1S_n, and T = U_n≥1T_n, we find geometric conditions on S or on T , such that S1 T1 implies that S ° CT for some constant C. For instance, under various symmetry conditions on S, we can show that C above can be chosen to equal d, the number of variables. We also show that C = d is sharp for a specific matrix convex set W^max(Bd) constructed from the unit ball Bd. This led us to find an essentially unique self-dual matrix convex set D, the self-dual matrix ball, for which corresponding inclusion and dilation results hold with constant C = d. For a certain class of polytopes, we obtain a considerable sharpening of such inclusion results involving polar duals. An illustrative example is that a sufficient condition for T to contain the free matrix cube e^(d)=U_n{(T₁,...,T_d)ϵM^d_n:≤} is that {x_j ≤ R^d :σ|x_j|≤1} ¹/_d T₁, that is, that ¹/_d T₁ contains the polar dual of the cube [.1, 1]d = C(d) 1 . 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 gnumber of variablesh d. There are also implications to the problem of existence of (unital) completely positive maps with prescribed values on a set of operators.