Facial structure of matrix convex sets

Abstract

This article investigates the notions of exposed points and (exposed) faces in the matrix convex setting. Matrix exposed points in finite dimensions were first defined by Kriel in 2019. Here this notion is extended to matrix convex sets in infinite-dimensional vector spaces. Then a connection between matrix exposed points and matrix extreme points is established: a matrix extreme point is ordinary exposed if and only if it is matrix exposed. This leads to a Krein-Milman type result for matrix exposed points that is due to Straszewicz-Klee in classical convexity: a compact matrix convex set is the closed matrix convex hull of its matrix exposed points. Several notions of a fixed-level as well as a multicomponent matrix face and matrix exposed face are introduced to extend the concepts of a matrix extreme point and a matrix exposed point, respectively. Their properties resemble those of (exposed) faces in the classical sense, e.g., it is shown that the $C^\ast$-extreme (matrix extreme) points of a matrix face (matrix multiface) of a matrix convex set $K$ are matrix extreme in $K$. As in the case of extreme points, any fixed-level matrix face is ordinary exposed if and only if it is a matrix exposed face. From this it follows that every fixed-level matrix face of a free spectrahedron is matrix exposed. On the other hand, matrix multifaces give rise to the noncommutative counterpart of the classical theory connecting (archimedean) faces of compact convex sets and (archimedean) order ideals of the corresponding function systems.