Operator systems and convex sets with many normal cones

TL;DR

This paper explores the convex geometric properties of operator systems, revealing how their structure leads to smoothness and clustering of exposed faces, with implications for the lattice of ground state projections.

Contribution

It demonstrates that operator systems have many normal cones, linking convex geometry with operator algebra, and minimizes assumptions needed for these properties.

Findings

Convex sets with many normal cones exhibit smoothness and face clustering.

Exposed faces are intersections of maximal exposed faces.

Results translate to the lattice of ground state projections.

Abstract

The state space of an operator system of $n$-by-$n$ matrices has, in a sense, many normal cones. Merely this convex geometrical property implies smoothness qualities and a clustering property of exposed faces. The latter holds since each exposed face is an intersection of maximal exposed faces. An isomorphism translates these results to the lattice of ground state projections of the operator system. We work on minimizing the assumptions under which a convex set has the mentioned properties.