Geodesic
The tracial Hahn–Banach theorem, polar duals, matrix convex sets, and projections of free spectrahedra
Journal of the European Mathematical Society, Tome 19 (2017) no. 6, pp. 1845-1897.

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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.
DOI : 10.4171/jems/707
Classification : 14-XX, 47-XX, 90-XX
Keywords: Linear matrix inequality (LMI), polar dual, LMI domain, spectrahedron, spectrahedrop, convex hull, free real algebraic geometry, noncommutative polynomial, cp interpolation, quantum channel, tracial hull, tracial Hahn–Banach theorem 
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     author = {J. William Helton and Igor Klep and Scott McCullough},
     title = {The tracial {Hahn{\textendash}Banach} theorem, polar duals, matrix convex sets, and projections of free spectrahedra},
     journal = {Journal of the European Mathematical Society},
     pages = {1845--1897},
     publisher = {mathdoc},
     volume = {19},
     number = {6},
     year = {2017},
     doi = {10.4171/jems/707},
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J. William Helton; Igor Klep; Scott McCullough. The tracial Hahn–Banach theorem, polar duals, matrix convex sets, and projections of free spectrahedra. Journal of the European Mathematical Society, Tome 19 (2017) no. 6, pp. 1845-1897. doi : 10.4171/jems/707. http://geodesic.mathdoc.fr/articles/10.4171/jems/707/

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