Skip to main content

Involutive operator algebras

Positivity volume 24, pages 13–53 (2020)Cite this article

Abstract

Examples of operator algebras with involution include the operator \(*\)-algebras occurring in noncommutative differential geometry studied recently by Mesland, Kaad, Lesch, and others, several classical function algebras, triangular matrix algebras, (complexifications) of real operator algebras, and an operator algebraic version of the complex symmetric operators studied by Garcia, Putinar, Wogen, Zhu, and others. We investigate the general theory of involutive operator algebras, and give many applications, such as a characterization of the symmetric operator algebras introduced in the early days of operator space theory.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

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

    Article  MathSciNet  Google Scholar 

  2. Arveson, W.B.: Analyticity in operator algebras. Am. J. Math. 89, 578–642 (1967)

    Article  MathSciNet  Google Scholar 

  3. Bearden, C.A., Blecher, D.P., Sharma, S.: On positivity and roots in operator algebras. Integral Equ. Oper. Theory 79, 555–566 (2014)

    Article  MathSciNet  Google Scholar 

  4. Bercovici, H.: Operator Theory and Arithmetic in \(H^\infty \), Mathematical Surveys and Monographs, 26. American Mathematical Society, Providence, RI (1988)

    Book  Google Scholar 

  5. Blackadar, B.: Operator Algebras. Theory of \(C^*\)-Algebras and von Neumann Algebras, Encyclopaedia of Mathematical Sciences, 122. Operator Algebras and Non-commutative Geometry, III. Springer-Verlag, Berlin (2006)

    Book  Google Scholar 

  6. Blecher, D.P.: Commutativity in operator algebras. Proc. Am. Math. Soc. 109, 709–715 (1990)

    Article  MathSciNet  Google Scholar 

  7. Blecher, D.P.: Noncommutative peak interpolation revisited. Bull. Lond. Math. Soc. 45, 1100–1106 (2013)

    Article  MathSciNet  Google Scholar 

  8. Blecher, D.P.: Generalization of C*-algebra methods via real positivity for operator and Banach algebras, In “Operator algebras and their applications: A tribute to Richard V. Kadison”. In: Doran, R.S., Park, E. (eds.) Contemporary Mathematics, vol. 671, pp. 35–66. American Mathematical Society, Providence, RI (2016)

    Google Scholar 

  9. Blecher, D.P., Hay, D.M., Neal, M.: Hereditary subalgebras of operator algebras. J. Oper. Theory 59, 333–357 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Blecher, D.P., Kaad, J., Mesland, B.: Operator \(\ast \)-correspondences in analysis and geometry. Proc. Lond. Math. Soc. 117, 303–344 (2018)

    Article  MathSciNet  Google Scholar 

  11. Blecher, D.P., Kirkpatrick, K., Neal, M., Werner, W.: Ordered involutive operator spaces. Positivity 11, 497–510 (2007)

    Article  MathSciNet  Google Scholar 

  12. Blecher, D.P., Le Merdy, C.: Operator Algebras and Their Modules–An Operator Space Approach. Oxford University Press, Oxford (2004)

    Book  Google Scholar 

  13. Blecher, D.P., Neal, M.: Open projections in operator algebras II: compact projections. Studia Math. 209, 203–224 (2012)

    Article  MathSciNet  Google Scholar 

  14. Blecher, D.P., Neal, M.: Completely contractive projections on operator algebras. Pac. J. Math. 283–2, 289–324 (2016)

    Article  MathSciNet  Google Scholar 

  15. Blecher, D.P., Read, C.J.: Operator algebras with contractive approximate identities. J. Funct. Anal. 261, 188–217 (2011)

    Article  MathSciNet  Google Scholar 

  16. Blecher, D.P., Read, C.J.: Operator algebras with contractive approximate identities II. J. Funct. Anal. 264, 1049–1067 (2013)

    Article  MathSciNet  Google Scholar 

  17. Blecher, D.P., Read, C.J.: Order theory and interpolation in operator algebras. Studia Math. 225, 61–95 (2014)

    Article  MathSciNet  Google Scholar 

  18. Blecher, D.P., Solel, B.: A double commutant theorem for operator algebras. J. Oper. Theory 51, 435–453 (2004)

    MathSciNet  MATH  Google Scholar 

  19. Blecher, D.P., Wang, Z.: Jordan operator algebras: basic theory. Mathematische Nachrichten 291, 1629–1654 (2018)

    Article  MathSciNet  Google Scholar 

  20. Blecher, D. P., Wang, Z.: Jordan operator algebras revisited, Preprint 2018, to appear, Mathematische Nachrichten

  21. Effros, E.G., Ruan, Z.: Operator Spaces. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  22. Garcia, S.R., Prodan, E., Putinar, M.: Mathematical and physical aspects of complex symmetric operators. J. Phys. A 47, 353001 (2014)

    Article  MathSciNet  Google Scholar 

  23. Garcia, S.R., Wogen, W.: Some new classes of complex symmetric operators. Trans. Amer. Math. Soc. 362, 6065–6077 (2010)

    Article  MathSciNet  Google Scholar 

  24. Guo, K., Ji, Y., Zhu, S.: A \(C^*\)-algebra approach to complex symmetric operators. Trans. Am. Math. Soc. 367, 6903–6942 (2015)

    Article  MathSciNet  Google Scholar 

  25. Hay, D.M.: Closed projections and peak interpolation for operator algebras. Integral Equ. Oper. Theory 57, 491–512 (2007)

    Article  MathSciNet  Google Scholar 

  26. Kaad, J., Lesch, M.: Spectral flow and the unbounded Kasparov product. Adv. Math. 248, 495–530 (2013)

    Article  MathSciNet  Google Scholar 

  27. Mesland, B.: Unbounded bivariant \(K\)-theory and correspondences in noncommutative geometry. J. Reine Angew. Math. 691, 101–172 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Nagy, B.Sz, Foias, C., Bercovici, H., Kérchy, L.: Harmonic Analysis of Operators on Hilbert Space, Second Edition. Revised and Enlarged Edition. Universitext. Springer, New York (2010)

    Book  Google Scholar 

  29. Pedersen, G.K.: Factorization in \(C^*\)-algebra. Exposition. Math. 16, 145–156 (1988)

    MathSciNet  MATH  Google Scholar 

  30. Peters, J., Wogen, W.: Commutative radical operator algebras. J. Oper. Theory 42, 405–424 (1999)

    MathSciNet  MATH  Google Scholar 

  31. Pisier, G.: Introduction to operator space theory, London Math. Soc. Lecture Note Series, 294, Cambridge University Press, Cambridge (2003)

  32. Sharma, S.: Real operator algebras and real completely isometric theory. Positivity 18, 95–118 (2014)

    Article  MathSciNet  Google Scholar 

  33. Shen, J., Zhu, S.: Complex symmetric generators for operator algebras. J. Oper. Theory 77, 421–454 (2017)

    Article  MathSciNet  Google Scholar 

  34. Wang, Z.: Theory of Jordan operator algebras and operator \(^*\)-algebras, PhD thesis University of Houston (2019)

  35. Zhu, S., Zhao, J.: Complex symmetric generators of \(C^*\)-algebras. J. Math. Anal. Appl. 456, 796–822 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This project grew out of [10], and we thank Jens Kaad and Bram Mesland for several ideas and perspectives learned there. We also thank Stephan Garcia–whose work on complex symmetric operators has influenced some results in our paper–for helpful conversations, and also Elias Katsoulis.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, TX, 77204-3008, USA

    David P. Blecher & Zhenhua Wang

Authors
  1. David P. Blecher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhenhua Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David P. Blecher.

Additional information

Publisher's Note

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

We acknowledge support by Simons Foundation grant 527078.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Blecher, D.P., Wang, Z. Involutive operator algebras. Positivity 24, 13–53 (2020). https://doi.org/10.1007/s11117-019-00664-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-019-00664-6

Keywords

  • Operator algebras
  • Involution
  • Accretive operator
  • Ideal
  • Hereditary subalgebra
  • Interpolation
  • Complex symmetric operator

Mathematics Subject Classification

  • Primary 46K50
  • 46L52
  • 47L07
  • 47L30
  • 47L75
  • Secondary 32T40
  • 46J15
  • 46L07
  • 46L85
  • 47B44
  • 47L25
  • 47L45