Complex free spectrahedra, absolute extreme points, and dilations

Evert and Helton proved that real free spectrahedra are the matrix convex hulls of their absolute extreme points. However, this result does not extend to complex spectrahedra, we examine multiple ways in which analogous can fail. We also develop some local techniques determine when sets (duals of) as part a continued study minimal maximal operator systems. These results apply both cases.