Geometric Aspects of $C^{\ast }$-Extreme Points

We provide a characterization of the $C^{\ast }$-extreme points of the closed unit ball of a von Neumann algebra and demonstrate that $C^{\ast }$-extremality is equivalent to both linear extremality and strong extremality. As an application, we characterize certain classes of von Neumann algebras in terms of their $C^{\ast }$-extreme points.