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Quantum symmetries of the twisted tensor products of C*-algebras

Jyotishman Bhowmick, Arnab Mandal, Sutanu Roy, Adam Skalski

Abstract

We consider the construction of twisted tensor products in the category of C*-algebras equipped with orthogonal filtrations and under certain assumptions on the form of the twist compute the corresponding quantum symmetry group, which turns out to be the generalised Drinfeld double of the quantum symmetry groups of the original filtrations. We show how these results apply to a wide class of crossed products of C*-algebras by actions of discrete groups. We also discuss an example where the hypothesis of our main theorem is not satisfied and the quantum symmetry group is not a generalised Drinfeld double.

Quantum symmetries of the twisted tensor products of C*-algebras

Abstract

We consider the construction of twisted tensor products in the category of C*-algebras equipped with orthogonal filtrations and under certain assumptions on the form of the twist compute the corresponding quantum symmetry group, which turns out to be the generalised Drinfeld double of the quantum symmetry groups of the original filtrations. We show how these results apply to a wide class of crossed products of C*-algebras by actions of discrete groups. We also discuss an example where the hypothesis of our main theorem is not satisfied and the quantum symmetry group is not a generalised Drinfeld double.

Paper Structure

This paper contains 19 sections, 29 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum groups
  4. Universal algebras of locally compact quantum groups
  5. Coactions, bicharacters and quantum group morphisms
  6. Twisted tensor product C*-algebras and the action of the Drinfeld double
  7. Universal $\textup{C}^*$-algebras of Drinfeld doubles and their universal property
  8. Coaction on the twisted tensor products
  9. Universal property of $\textup{C}_0^\textup{u}(\mathfrak{D}_{\textup{V}})$
  10. Orthogonal filtrations of $C^*$-algebras and their quantum symmetries
  11. The quantum symmetry group of an orthogonal filtration
  12. Orthogonal filtration of a twisted tensor product
  13. Quantum symmetries of twisted tensor product
  14. The case of the reduced crossed products
  15. Quantum symmetries of reduced crossed products
  16. ...and 4 more sections

Key Result

Lemma \oldthetheorem

Let $A, B, C$ be $\textup{C}^*$-algebras with $A\subseteq\mathcal{M}(D)$ and let $\gamma\colon A\to\mathcal{M}(D\otimes C)$ be a $^*$-homomorphism such that $\gamma(A)(1_{D}\otimes C)=A\otimes C$. Then $\gamma\in\textup{Mor}(A,A\otimes C)$.

Theorems & Definitions (71)

  • Definition \oldthetheorem
  • Example \oldthetheorem
  • Example \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • Definition \oldthetheorem
  • Example \oldthetheorem
  • ...and 61 more