The Hahn-Banach Separation Theorem in Free Convexity

Abstract

This thesis presents and develops theorems on separation in the context of free convexity and matrix convex sets. After presenting a proof of the Hahn-Banach Separation Theorem, the main work in this thesis is a treatment of the Effros-Winkler Hahn-Banach Separation Theorem in the setting of matrix convex sets. This result is subsequently interpreted as a theorem concerning noncommutative sets in free analysis. Lastly, a new and complete characterization (in terms of liner pencils) of the matrix convex hull of a g-tuple of complex matrices is established.