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

Operational extreme points and Cuntz's canonical endomorphism