Hann-Banach-Arveson extension theorem and Kadison isomorphism

Anilesh Mohari

Hann-Banach-Arveson extension theorem and Kadison isomorphism

Operator Algebras 2018-08-28 v9

Authors: Anilesh Mohari

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.

Keywords

Hahn-Banach-Arveson extension theorem Kadison isomorphism operator systems C*-algebras complete order isomorphism completely positive maps cocycle conjugacy matrix algebras

Cite

@article{arxiv.1304.6849,
  title  = {Hann-Banach-Arveson extension theorem and Kadison isomorphism},
  author = {Anilesh Mohari},
  journal= {arXiv preprint arXiv:1304.6849},
  year   = {2018}
}

Comments

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

Hann-Banach-Arveson extension theorem and Kadison isomorphism

Abstract

Keywords

Cite

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