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This article investigates matrix convex sets and introduces their tracialanalogs which we call contractively tracial convex sets. In both contextscompletely positive (cp) maps play a central role: unital cp maps in the caseof matrix convex sets and trace preserving cp (CPTP) maps in the case ofcontractively 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) = A_0 + A_1 x_1 + ... + A_g x_g > 0 and their matrix convex solution sets{ X : L(X) is positive semidefinite }, called free spectrahedra. TheEffros-Winkler Hahn-Banach Separation Theorem for matrix convex sets statesthat matrix convex sets are solution sets of LMIs with operator coefficients.Motivated in part by cp interpolation problems, we develop the foundations ofconvex analysis and duality in the tracial setting, including tracial analogsof the Effros-Winkler Theorem.  The projection of a free spectrahedron in g+h variables to g variables is amatrix convex set called a free spectrahedrop. As a class, free spectrahedropsare more general than free spectrahedra, but at the same time more tractablethan general matrix convex sets. Moreover, many matrix convex sets can beapproximated from above by free spectrahedrops. Here a number of fundamentalresults for spectrahedrops and their polar duals are established. For example,the free polar dual of a free spectrahedrop is again a free spectrahedrop. Wealso give a Positivstellensatz for free polynomials that are positive on a freespectrahedrop.

The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra