A Radon-Nikodym theorem for completely n-positive linear maps on pro-C*-algebras and its applications

TL;DR

This paper extends the Radon-Nikodym theorem to the setting of completely n-positive linear maps on pro-C*-algebras, providing characterizations of order relations, pure elements, and extreme points via associated representations.

Contribution

It introduces a Radon-Nikodym theorem for completely n-positive maps on pro-C*-algebras and characterizes pure and extreme points through their induced representations.

Findings

Order relation characterized via associated representations

Pure elements characterized in terms of representations

Extreme points characterized through induced representations

Abstract

The order relation on the set of completely n-positive linear maps from a pro-C*-algebra A to L(H), the C*-algebra of bounded linear operators on a Hilbert space H, is characterized in terms of the representation associated with each completely n-positive linear map. Also, the pure elements in the set of all completely n-positive linear maps from A to L(H) and the extreme points in the set of unital completely n-positive linear maps from A to L(H) are characterized in terms of the representation induced by each completely n-positive linear map.