The Krein-Milman theorem in operator convexity

Abstract

We generalize the Krein-Milman theorem to the setting of matrix convex sets of Effros-Winkler, extending the work of Farenick-Morenz on compact C ∗ ^* -convex sets of complex matrices and the matrix state spaces of C ∗ ^* -algebras. An essential ingredient is to prove the non-commutative analogue of the fact that a compact convex set K K may be thought of as the state space of the space of continuous affine functions on K K .