Operational extreme points and Cuntz's canonical endomorphism

Abstract

Based on the fact that the Cuntz algebra $O_n$ is generated by the operators consisting of a finite operatorional partition, we study the notion of operational extreme points (which we introduce here) by using several completely positive maps on $O_n$. As a typical example, we show that the Cuntz's canonical endomorphism $Phi_n$ is an operational extreme point in the set of completely positive maps on $O_n$ and that it induces a completely positive map which is extreme but not operational extreme, etc.