Hann-Banach-Arveson extension theorem and Kadison isomorphism

Abstract

Let $C^*(\cls)$ be the $C^{\ast }$ algebra generated by an operator system $\cls$ i.e. a unital $\ast$-closed subspace of a unital $C^{\ast }$ 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 $C^{\ast }$-isomorphism $\cli:C^*(\cls) \raro C^*(\cls')$. However, the same statement with more general operator systems of infinite dimensional $C^{\ast }$-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.

The paper needs further correction and stated main theorem is false unless we add some more structures to it! Such a generalisation is finally aimed to classify extremal elements of unital completely positive map