Complex Free Spectrahedra, Absolute Extreme Points, and Dilations

TL;DR

This paper investigates the differences between real and complex free spectrahedra, exploring conditions under which certain convex hull properties hold or fail, and develops techniques to identify when matrix convex sets are not free spectrahedra.

Contribution

It extends the study of matrix convex sets by analyzing the failure of real spectrahedra properties in the complex case and introduces local techniques to identify non-spectrahedral convex sets.

Findings

Real free spectrahedra are matrix convex hulls of their absolute extreme points.

The analogous property does not hold for complex free spectrahedra.

Developed local techniques to determine when matrix convex sets are not free spectrahedra.

Abstract

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 free spectrahedra, and we examine multiple ways in which the analogous result can fail. We also develop some local techniques to determine when matrix convex sets are not (duals of) free spectrahedra, as part of a continued study of minimal and maximal matrix convex sets and operator systems. These results apply to both the real and complex cases.