THE TRACIAL HAHN-BANACH THEOREM, POLAR DUALS, MATRIX CONVEX SETS, AND PROJECTIONS OF FREE SPECTRAHEDRA
J. William Helton, Igor Klep, Scott McCullough · 2016
Abstract. This article investigates matrix convex sets. It introduces tracial analogs, which we call contractively tracial convex sets. Critical in both contexts are completely positive (cp) maps. While unital cp maps tie into matrix convex sets, trace preserving cp (CPTP) maps tie into contractively tracial sets. CPTP maps are sometimes called quantum channels and are central to quantum information theory. Free convexity is intimately connected with Linear Matrix Inequalities (LMIs) L(x) = A0 + A1x1 + · · ·+ Agxg 0 and their matrix convex solution sets {X: L(X) 0}, called free spectrahedra. The classical Effros-Winkler theorem states that matrix convex sets are solution sets of LMIs with operator coefficients. Indeed, this is the Hahn-Banach theorem for matrix convex sets. We give the analogous theorem for contractively tracial sets. The projection of a free spectrahedron in g+h variables to g variables is a matrix convex set called a free spectrahedrop. As a class, free spectrahedrops are more general than free spectrahedra, but at the same time more tractable than general matrix convex sets. More-over, many matrix convex sets can be approximated from above by free spectrahedrops. We give a number of results on free spectrahedrops and their polar duals. For example, the free polar dual of a free spectrahedrop is a free spectrahedrop. We also give a Positivstellensatz for free polynomials which are positive on a free spectrahedrop. 1.