Duals of quantum semigroups with involution

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.

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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.

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Correspondence to Yulia Kuznetsova.

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Communicated by Uwe Franz.

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

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Keywords

  • Quantum semigroups
  • Locally compact quantum groups
  • Pontryagin duality

Mathematics Subject Classification

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