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Hyperrigid generators in \(C*\)-algebras

  • P. ShankarEmail author
Original Research Paper
  • 1 Downloads

Abstract

In this article, we show that, if \(S\in \mathcal {B}(H)\) is irreducible and essential unitary, then \(\{S,SS^*\}\) is a hyperrigid generator for the unital \(C^*\)-algebra \(\mathcal {T}\) generated by S. We prove that, if T is an operator in \(\mathcal {B}(H)\) that generates a unital \(C^*\)-algebra \(\mathcal {A}\) then \(\{T,T^*T,TT^*\}\) is a hyperrigid generator for \(\mathcal {A}\). As a corollary it follows that, if \(T\in \mathcal {B}(H)\) is normal then \(\{T,TT^*\}\) is hyperrigid generator for the unital \(C^*\)-algebra generated by T and if \(T\in \mathcal {B}(H)\) is unitary then \(\{T\}\) is hyperrigid generator for the \(C^*\)-algebra generated by T. We show that if \(V\in \mathcal {B}(H)\) is an isometry (not unitary) that generates the \(C^*\)-algebra \(\mathcal {A}\) then the minimal generating set \(\{V\}\) is not hyperrigid for \(\mathcal {A}\).

Keywords

Hyperrigidity Essential unitary operator Unital completely positive map Unique extension property 

Mathematics Subject Classification

46L07 46L52 47A13 47L80 

Notes

Acknowledgements

The author would like to thank Orr Moshe Shalit for valuable discussions and for a careful reading of this manuscript and some constructive comments. The author would like to thank Douglas Farenick and B. V. Rajarama Bhat for valuable discussions. The author would like to thank Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore, India for providing visiting scientist post doctoral fellowship. The author would like to thank Dr. Dhriti Ranjan Dolai, INSPIRE faculty, Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore for partial support of visiting scientist post doctoral fellowship from DST/INSPIRE/04/2017/000109 INSPIRE Grant. The author is very thankful to the referees for pointing out some errors and giving suggestions for improvement of the presentation of the article.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no confict of interest.

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Copyright information

© Forum D'Analystes, Chennai 2019

Authors and Affiliations

  1. 1.Statistics and Mathematics UnitIndian Statistical InstituteBangaloreIndia

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