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The maximal quantum group-twisted tensor product of C*-algebras

Sutanu Roy, Thomas Timmermann

Abstract

We construct a maximal counterpart to the minimal quantum group-twisted tensor product of $C^{*}$-algebras studied by Meyer, Roy and Woronowicz, which is universal with respect to representations satisfying braided commutation relations. Much like the minimal one, this product yields a monoidal structure on the coactions of a quasi-triangular $C^{*}$-quantum group, the horizontal composition in a bicategory of Yetter-Drinfeld $C^{*}$-algebras, and coincides with a Rieffel deformation of the non-twisted tensor product in the case of group coactions.

The maximal quantum group-twisted tensor product of C*-algebras

Abstract

We construct a maximal counterpart to the minimal quantum group-twisted tensor product of -algebras studied by Meyer, Roy and Woronowicz, which is universal with respect to representations satisfying braided commutation relations. Much like the minimal one, this product yields a monoidal structure on the coactions of a quasi-triangular -quantum group, the horizontal composition in a bicategory of Yetter-Drinfeld -algebras, and coincides with a Rieffel deformation of the non-twisted tensor product in the case of group coactions.

Paper Structure

This paper contains 22 sections, 53 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\mathrm{C}^*$-bialgebras
  4. $\mathrm{C}^*$-quantum groups Baaj-Skandalis:UnitairesSoltan-Woronowicz:Remark_manageableSoltan-Woronowicz:Multiplicative_unitariesWoronowicz:Mult_unit_to_Qgrp
  5. Universal quantum groups Soltan-Woronowicz:Multiplicative_unitariesMeyer-Roy-Woronowicz:Homomorphisms
  6. Morphisms of quantum groups Meyer-Roy-Woronowicz:Homomorphisms
  7. Coactions of $\mathrm{C}^{*}$-quantum groups
  8. Braided commutation relations
  9. Braided-commuting representations
  10. The tensor product
  11. The cubical tricategory
  12. Reduced braided-commuting representations
  13. The maximal twisted tensor product of $\mathrm{C}^{*}$-algebras
  14. Some properties and special cases of the maximal twisted tensor product
  15. Exactness
  16. ...and 7 more sections

Key Result

Theorem \oldthetheorem

Let $\mathcal{H}$ be a separable Hilbert space and $\mathbb{W}\in\mathcal{U}(\mathcal{H}\otimes\mathcal{H})$ a modular multiplicative unitary.

Theorems & Definitions (128)

  • Definition \oldthetheorem: Baaj-Skandalis:Unitaires*Definition 1.1
  • Theorem \oldthetheorem: Soltan-Woronowicz:Remark_manageableSoltan-Woronowicz:Multiplicative_unitariesWoronowicz:Mult_unit_to_Qgrp
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • ...and 118 more