pp 1–41 | Cite as

Involutive operator algebras

  • David P. BlecherEmail author
  • Zhenhua Wang


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.


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 



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.


  1. 1.
    Arveson, W.B.: Subalgebras of \(C^{*}\)-algebras. Acta Math. 123, 141–224 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arveson, W.B.: Analyticity in operator algebras. Am. J. Math. 89, 578–642 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bearden, C.A., Blecher, D.P., Sharma, S.: On positivity and roots in operator algebras. Integral Equ. Oper. Theory 79, 555–566 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bercovici, H.: Operator Theory and Arithmetic in \(H^\infty \), Mathematical Surveys and Monographs, 26. American Mathematical Society, Providence, RI (1988)Google Scholar
  5. 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)zbMATHGoogle Scholar
  6. 6.
    Blecher, D.P.: Commutativity in operator algebras. Proc. Am. Math. Soc. 109, 709–715 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Blecher, D.P.: Noncommutative peak interpolation revisited. Bull. Lond. Math. Soc. 45, 1100–1106 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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. 9.
    Blecher, D.P., Hay, D.M., Neal, M.: Hereditary subalgebras of operator algebras. J. Oper. Theory 59, 333–357 (2008)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Blecher, D.P., Kaad, J., Mesland, B.: Operator \(\ast \)-correspondences in analysis and geometry. Proc. Lond. Math. Soc. 117, 303–344 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Blecher, D.P., Kirkpatrick, K., Neal, M., Werner, W.: Ordered involutive operator spaces. Positivity 11, 497–510 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Blecher, D.P., Le Merdy, C.: Operator Algebras and Their Modules–An Operator Space Approach. Oxford University Press, Oxford (2004)CrossRefzbMATHGoogle Scholar
  13. 13.
    Blecher, D.P., Neal, M.: Open projections in operator algebras II: compact projections. Studia Math. 209, 203–224 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Blecher, D.P., Neal, M.: Completely contractive projections on operator algebras. Pac. J. Math. 283–2, 289–324 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Blecher, D.P., Read, C.J.: Operator algebras with contractive approximate identities. J. Funct. Anal. 261, 188–217 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Blecher, D.P., Read, C.J.: Operator algebras with contractive approximate identities II. J. Funct. Anal. 264, 1049–1067 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Blecher, D.P., Read, C.J.: Order theory and interpolation in operator algebras. Studia Math. 225, 61–95 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Blecher, D.P., Solel, B.: A double commutant theorem for operator algebras. J. Oper. Theory 51, 435–453 (2004)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Blecher, D.P., Wang, Z.: Jordan operator algebras: basic theory. Mathematische Nachrichten 291, 1629–1654 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Blecher, D. P., Wang, Z.: Jordan operator algebras revisited, Preprint 2018, to appear, Mathematische NachrichtenGoogle Scholar
  21. 21.
    Effros, E.G., Ruan, Z.: Operator Spaces. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar
  22. 22.
    Garcia, S.R., Prodan, E., Putinar, M.: Mathematical and physical aspects of complex symmetric operators. J. Phys. A 47, 353001 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Garcia, S.R., Wogen, W.: Some new classes of complex symmetric operators. Trans. Amer. Math. Soc. 362, 6065–6077 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Guo, K., Ji, Y., Zhu, S.: A \(C^*\)-algebra approach to complex symmetric operators. Trans. Am. Math. Soc. 367, 6903–6942 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hay, D.M.: Closed projections and peak interpolation for operator algebras. Integral Equ. Oper. Theory 57, 491–512 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kaad, J., Lesch, M.: Spectral flow and the unbounded Kasparov product. Adv. Math. 248, 495–530 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mesland, B.: Unbounded bivariant \(K\)-theory and correspondences in noncommutative geometry. J. Reine Angew. Math. 691, 101–172 (2014)MathSciNetzbMATHGoogle Scholar
  28. 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)CrossRefGoogle Scholar
  29. 29.
    Pedersen, G.K.: Factorization in \(C^*\)-algebra. Exposition. Math. 16, 145–156 (1988)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Peters, J., Wogen, W.: Commutative radical operator algebras. J. Oper. Theory 42, 405–424 (1999)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Pisier, G.: Introduction to operator space theory, London Math. Soc. Lecture Note Series, 294, Cambridge University Press, Cambridge (2003)Google Scholar
  32. 32.
    Sharma, S.: Real operator algebras and real completely isometric theory. Positivity 18, 95–118 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Shen, J., Zhu, S.: Complex symmetric generators for operator algebras. J. Oper. Theory 77, 421–454 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wang, Z.: Theory of Jordan operator algebras and operator \(^*\)-algebras, PhD thesis University of Houston (2019)Google Scholar
  35. 35.
    Zhu, S., Zhao, J.: Complex symmetric generators of \(C^*\)-algebras. J. Math. Anal. Appl. 456, 796–822 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA

Personalised recommendations