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- Title
The tracial Hahn--Banach theorem, polar duals, matrix convex sets, and projections of free spectrahedra.
- Authors
Helton, J. William; Klep, Igor; McCullough, Scott
- Abstract
This article investigates matrix convex sets and introduces their tracial analogs which we call contractively tracial convex sets. In both contexts completely positive (cp) maps play a central role: unital cp maps in the case of matrix convex sets and trace preserving cp (CPTP) maps in the case of contractively tracial convex sets. CPTP maps, also known as quantum channels, are fundamental objects in quantum information theory. Free convexity is intimately connected with Linear Matrix Inequalities (LMIs) L(x) = A0 + A/x/ + ... Agxg ≥ 0 and their matrix convex solution sets (X : L(X) ≥ 0), called free spectrahedra. The Effros--Winkler Hahn-Banach Separation Theorem for matrix convex sets states that matrix convex sets are solution sets of LMIs with operator coefficients. Motivated in part by cp interpolation problems, we develop the foundations of convex analysis and duality in the tracial setting, including tracial analogs of the Effros--Winkler Theorem. The projection of a free spectrahedron in g C 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.Moreover, many matrix convex sets can be approximated from above by free spectrahedrops. Here a number of fundamental results for spectrahedrops and their polar duals are established. For example, the free polar dual of a free spectrahedrop is again a free spectrahedrop. We also give a Positivstellensatz for free polynomials that are positive on a free spectrahedrop.
- Subjects
HAHN-Banach theorem; MATHEMATICS; CONVEX sets; CONVEX domains; DIFFEOMORPHISMS
- Publication
Journal of the European Mathematical Society (EMS Publishing), 2017, Vol 19, Issue 6, p1845
- ISSN
1435-9855
- Publication type
Article
- DOI
10.4171/JEMS/707