Duals of quantum semigroups with involution

Advances in Operator Theory volume 5, pages167–203(2020)Cite this article

Abstract

We define a category \({\mathcal {QSI}}\) of quantum semigroups with involution and a duality map on it. For locally compact quantum groups, this establishes a duality between the universal algebras of Kustermans. Objects in \({\mathcal {QSI}}\) are von Neumann algebras with only comultiplication and coinvolution (antipode). Other examples are also given.

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

Access options

Buy single article

Instant access to the full article PDF.

US$ 39.95

Tax calculation will be finalised during checkout.

References

  1. 1.

    Banica, T., Skalski, A.: The quantum algebra of partial Hadamard matrices. Linear Algorithm Appl. 469, 364–380 (2015)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Cuntz, J., Echterhoff, S., Li, X.: On the K-theory of the \(C^*\)-algebra generated by the left regular representation of an Ore semigroup. J. Eur. Math. Soc. 17(3), 645–687 (2015)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Das, B., Mrozinski, C.: On a quantum version of Ellis joint continuity theorem. Illinois J. Math. 59(4), 839–858 (2015)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Dixmier, J.: \(C^*\)-algebras. North Holland (1977)

  5. 5.

    Enock, M., Schwartz, J.-M.: Kac algebras and duality of locally compact groups. Springer, Berlin (1992)

    Google Scholar 

  6. 6.

    Henrichs, R.: Decomposition of invariant states and nonseparable \(C^*\)-algebras. Publ. Res. Inst. Math. Sci. 18, 159–181 (1982)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kirchberg, E.: Darstellungen coinvolutiver Hopf-\(W^*\)-Algebren und ihre Anwendung in der nicht-abelschen Dualitätstheorie lokalkompakter Gruppen. Thesis, Berlin (1977)

  8. 8.

    Kustermans, J.: Locally compact quantum groups in the universal setting. Int. J. Math. 12, 289 (2001)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Kustermans, J., Vaes, S.: Locally compact quantum groups. Annales ENS 4ème série 33(6), 837–934 (2000)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Kustermans, J., Vaes, S.: Locally compact quantum groups in the von Neumann algebraic setting. Math. Scand. 92, 68–92 (2003)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Kuznetsova, Yu.: A duality of locally compact groups that does not involve the Haar measure. Math. Scand. 116, 250–286 (2015)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Li, X.: Semigroup C*-algebras and amenability of semigroups. J. Funct. Anal. 262(10), 4302–4340 (2012)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Palmer, ThW: Banach algebras and the general theory of *-algebras, vol. II. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  14. 14.

    Ruppert, W.: Compact semitopological semigroups: an intrinsic theory. Lecture Notes Math. 1079, Springer, Berlin (1984)

    Google Scholar 

  15. 15.

    Sołtan, P.M., Woronowicz, S.L.: From multiplicative unitaries to quantum groups II. J. Funct. Anal. 252(1), 42–67 (2007)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Timmermann, Th.: An invitation to quantum groups and duality. EMS (2008)

  17. 17.

    Wils, W.: Direct integrals of Hilbert spaces I, II. Math. Scand. 26, 73–88 and 89–102 (1970)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Woronowicz, S.L.: From multiplicative unitaries to quantum groups. Int. J. Math. 7(1), 127–149 (1996)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

This work was partially supported by the Simons Foundation grant 346300 and the Polish Government MNiSW 2015-2019 matching fund. The author was also supported by the French “Investissements d’Avenir” program, project ISITE-BFC (contract ANR-15-IDEX-03) and the travel grant PHC Star 2016 36618SE of the French Ministry of Foreign Affairs.

Author information

Affiliations

  1. University Bourgogne Franche-Comté, 16 route de Gray, 25030, Besançon, France

    Yulia Kuznetsova

Authors
  1. Yulia Kuznetsova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yulia Kuznetsova.

Additional information

Communicated by Uwe Franz.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kuznetsova, Y. Duals of quantum semigroups with involution. Adv. Oper. Theory 5, 167–203 (2020). https://doi.org/10.1007/s43036-019-00011-2

Download citation

Keywords

  • Quantum semigroups
  • Locally compact quantum groups
  • Pontryagin duality

Mathematics Subject Classification

  • 22D35
  • 22D20
  • 22D25
  • 43A10
  • 16T10