Operational extreme points and Cuntz's canonical endomorphism

TL;DR

This paper introduces the concept of operational extreme points in the context of Cuntz algebras and demonstrates that Cuntz's canonical endomorphism is an example of such an extreme point, revealing new structural insights.

Contribution

It defines operational extreme points using completely positive maps on Cuntz algebras and shows that the canonical endomorphism exemplifies this concept, highlighting its unique properties.

Findings

Cuntz's canonical endomorphism is an operational extreme point.

The canonical endomorphism induces an extreme but not operational extreme map.

Operational extreme points provide new structural understanding of Cuntz algebras.

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.