Hyperrigid generators in \(C*\)-algebras

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 \(\mathcal {A}\).

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Agler, J. 1982. The Arveson extension theorem and coanalytic models. Integral Equations Operator Theory 5 (5): 608–631.

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Arveson, W.B. 1969. Subalgebras of \(C^{\ast } \)-algebras. Acta Mathematica 123: 141–224.

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Arveson, W.B. 1972. Subalgebras of \(C^{\ast } \)-algebras, II. Acta Mathematica 128 (3–4): 271–308.

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Arveson, W.B. 2011. The noncommutative Choquet boundary II: Hyperrigidity. Israel Journal of Mathematics 184: 349–385.

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Brown, L., Douglas, R., and Fillmore, P. 1973. Unitary equivalence modulo the compact operators and extensions of C*-algebras. In Proceedings of Conference on Operator Theory, Halifax, NS, Lecture Notes in Mathematics, 3445. Berlin: Springer.

  6. 6.

    Clouatre, R. 2018. Unperforated pairs of operator spaces and hyperrigidity of operator systems. Canadian Journal of Mathematics 70 (6): 1236–1260.

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Clouatre, R., and M. Hartz. 2018. Multiplier algebras of complete Nevanlinna–Pick spaces: dilations, boundary representations and hyperrigidity. Journal of Functional Analysis 274 (6): 1690–1738.

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Davidson, K.R., and Kennedy, M. Choquet order and hyperrigidity for function systems. arXiv:1608.02334v1.

  9. 9.

    Dor-On, A., and G. Salomon. 2018. Full Cuntz–Krieger dilations via non-commutative boundaries. Journal of the London Mathematical Society (2) 98 (2): 416–438.

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Katsoulis, E., and Ramsey, C. The hyperrigidity of tensor algebras of \(C^*\)-correspondences. arXiv:1905.10332v2.

  11. 11.

    Kennedy, M., and O.M. Shalit. 2015. Essential normality, essential norms and hyperrigidity. Journal of Functional Analysis 268 (10): 2990–3016.

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Klesky, C. 2014. Korovkin-type properties for completely positive maps. Illinois Journal of Mathematics 58 (4): 1107–1116.

    MathSciNet  Article  Google Scholar 

  13. 13.

    Korovkin, P.P. 1960. Linear operators and approximation theory. Delhi: Hindustan Publishing Corp.

    Google Scholar 

  14. 14.

    Namboodiri, M.N.N., S. Pramod, P. Shankar, and A.K. Vijayarajan. 2018. Quasi hyperrigidity and weak peak points for non commutative operator systems. Proceedings of the Indian Academy of Sciences: Mathematical Sciences 128 (5): 128:66.

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Richter, S., and C. Sunberg. 2010. Joint extensions in families of contractive commuting operator tuples. Journal of Functional Analysis 258 (10): 3319–3346.

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Saskin, Y.A. 1966. Korovkin systems in spaces of continuous functions. American Mathematical Society Translations 54 (2): 125–144.

    Article  Google Scholar 

  17. 17.

    Salomon, G. 2019. Hyperrigid subsets of graph \(C^*\)-algebras and the property of rigidity at zero. Journal of Operator Theory 81 (1): 61–79.

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Shankar, P., and A.K. Vijayarajan. 2018. Tensor products of hyperrigid operator systems. Annals of Functional Analysis 9 (3): 369–375.

    MathSciNet  MATH  Article  Google Scholar 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to P. Shankar.

Ethics declarations

Conflict of interest

The authors declare that they have no confict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shankar, P. Hyperrigid generators in \(C*\)-algebras. J Anal 28, 791–797 (2020). https://doi.org/10.1007/s41478-019-00199-9

Download citation

Keywords

  • Hyperrigidity
  • Essential unitary operator
  • Unital completely positive map
  • Unique extension property

Mathematics Subject Classification

  • 46L07
  • 46L52
  • 47A13
  • 47L80