### Topics in Operator Theory, Lecture 8: matrix convexity

In this lecture we will encounter the notion of matrix convexity. Matrix convexity is an active area of research today, and an important tool in noncommutative analysis. We will define *matrix convex sets,* and we will see that closed matrix convex sets have *matrix extreme points* which play a role similar to extreme points in analysis. As an example of a matrix convex set, we will study the set of all *matrix states*. We will use these notions to outline the proof that there are sufficiently many pure UCP maps, something that was left open from the previous lecture.