Convex geometry of completely positive maps

A linear map between matrix spaces is called completely positive (cp) if it preserves positive semidefiniteness, even when ampliated to higher-dimensional spaces. In this talk we demonstrate how cp maps are closely related to a particular class of convex sets: the solution sets of linear matrix inequalities (LMIs), known as spectrahedra.
Spectrahedra generalize polyhedra and have gained prominence since the 1990s due to their central role in semidefinite programming (SDP). The main result will explain how a cp map gives rise to a linear Positivstellensatz certificate and vice-versa.
The talk is based on joint works with Bill Helton and Scott McCullough.