Duality, extreme points and hulls for noncommutative partial convexity

This article studies generalizations of (matrix) convexity, including partial convexity and biconvexity, under the umbrella of $\Gamma$-convexity. Here $\Gamma$ is a tuple of free symmetric polynomials determining the geometry of a $\Gamma$-convex set. The paper introduces the notions of $\Gamma$-operator systems and $\Gamma$-ucp maps and establishes a Webster-Winkler type categorical duality between $\Gamma$-operator systems and $\Gamma$-convex sets. Next, a notion of an extreme point for $\Gamma$-convex sets is defined, paralleling the concept of a free extreme point for a matrix convex set. To ensure the existence of such points, the matricial sets considered are extended to include an operator level. It is shown that the $\Gamma$-extreme points of an operator $\Gamma$-convex set $K$ are in correspondence with the free extreme points of the operator convex hull of $\Gamma(K).$ From this result, a Krein-Milman theorem for $\Gamma$-convex sets follows. Finally, relying on the results of Helton and the first two authors, a construction of an approximation scheme for the $\Gamma$-convex hull of the matricial positivity domain {(also known as a free semialgebraic set)} $D_p$ of a free symmetric polynomial $p$ is given. The approximation consists of a decreasing family of $\Gamma$-analogs of free spectrahedra, whose projections, under mild assumptions, in the limit yield the $\Gamma$-convex hull of $D_p.$