Hann-Banach-Arveson extension theorem and Kadison isomorphism

TL;DR

This paper proves a unique extension property for complete order isomorphisms between operator systems in matrix algebras, with applications to characterizing extreme points of unital completely positive maps.

Contribution

It establishes a new extension theorem for operator systems in matrix algebras and characterizes extreme points of unital CP maps up to cocycle conjugacy.

Findings

Complete order isomorphisms extend uniquely to $C^*$-isomorphisms in matrix algebras.

The extension property fails for infinite-dimensional $C^*$-algebras.

Characterization of extreme points of unital CP maps up to cocycle conjugacy.

Abstract

Let $C^*(\cls)$ be the $C^*$ algebra generated by an operator system $\cls$ i.e. a unital $*$-closed subspace of a unital $C^*$ algebra $\cla$. We prove that any complete order isomorphism $\cli:\cls \raro \cls'$ between two such operator systems of matrix algebras has a unique extension to a $C^*$-isomorphism $\cli:C^*(\cls) \raro C^*(\cls')$. However, the same statement with more general operator systems of infinite dimensional $C^*$-algebra is false. As an application of this result, we characterise upto cocycle conjugacy the extreme points of unital completely positive maps on matrix algebra.