Facial structure of matrix convex sets

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