Matrix Convex Sets Without Absolute Extreme Points

TL;DR

This paper demonstrates that certain closed bounded matrix convex sets, constructed from tuples of compact operators with no finite dimensional reducing subspaces, lack absolute extreme points, challenging previous notions of minimal spanning sets.

Contribution

It introduces a class of matrix convex sets without absolute extreme points, providing new insights into the structure of matrix convexity and extreme points.

Findings

Existence of matrix convex sets without absolute extreme points

Construction using tuples of compact operators with no finite dimensional reducing subspaces

Challenges assumptions about minimal spanning sets in matrix convexity

Abstract

This article shows the existence of a class of closed bounded matrix convex sets which do not have absolute extreme points. The sets we consider are noncommutative sets, $K_X$, formed by taking matrix convex combinations of a single tuple $X$. In the case that $X$ is a tuple of compact operators with no nontrivial finite dimensional reducing subspaces, $K_X$ is a closed bounded matrix convex set with no absolute extreme points. A central goal in the theory of matrix convexity is to find a natural notion of an extreme point in the dimension free setting which is minimal with respect to spanning. Matrix extreme points are the strongest type of extreme point known to span matrix convex sets; however, they are not necessarily the smallest set which does so. Absolute extreme points, a more restricted type of extreme points that are closely related to Arveson's boundary, enjoy a strong notion of minimality should they span. This result shows that matrix convex sets may fail to be spanned by their absolute extreme points.