$C^ \ast$-extreme points of some compact $C^ \ast$-convex sets

Abstract:

In the ${C^{\ast }}$-algebra ${M_n}$ of complex $n \times n$ matrices, we consider the notion of noncommutative convexity called ${C^{\ast }}$-convexity and the corresponding notion of a ${C^{\ast }}$-extreme point. We prove that each irreducible element of ${M_n}$ is a ${C^{\ast }}$-extreme point of the ${C^{\ast }}$-convex set it generates, and we classify the ${C^{\ast }}$-extreme points of any ${C^{\ast }}$-convex set generated by a compact set of normal matrices.