Geometric Aspects of $C^*$-Extreme Points

TL;DR

This paper characterizes the $C^*$-extreme points of the unit ball in von Neumann algebras, showing their equivalence to linear and strong extremality, and applies this to classify certain von Neumann algebras.

Contribution

It provides a new characterization of $C^*$-extreme points and links them to other forms of extremality within von Neumann algebras, offering insights into their structure.

Findings

$C^*$-extreme points are equivalent to linear and strong extremality

Characterization of von Neumann algebras via $C^*$-extreme points

Applications to classifying specific von Neumann algebras

Abstract

We provide a characterization of the $C^*$-extreme points of the closed unit ball of a von Neumann algebra and demonstrate that $C^*$-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 $C^*$-extreme points.