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Faithful actions of locally compact quantum groups on classical spaces

Debashish Goswami, Sutanu Roy

Abstract

It is well-known that no non-Kac compact quantum group can faithfully act on $C(X)$ for a classical, compact Hausdorff space $X$. However, in this article we show that this is no longer true if we go to non-compact spaces and non-compact quantum groups, by exhibiting a large class of examples of locally compact quantum groups coming from bicrossed product construction, including non-Kac ones, which can faithfully and ergodically act on classical (non-compact) spaces. However, none of these actions can be isometric in the sense of Goswami, leading to the conjecture that the result obtained by Goswami and Joardar about non-existence of genuine quantum isometry of classical compact connected Riemannian manifolds may hold in the non-compact case as well.

Faithful actions of locally compact quantum groups on classical spaces

Abstract

It is well-known that no non-Kac compact quantum group can faithfully act on for a classical, compact Hausdorff space . However, in this article we show that this is no longer true if we go to non-compact spaces and non-compact quantum groups, by exhibiting a large class of examples of locally compact quantum groups coming from bicrossed product construction, including non-Kac ones, which can faithfully and ergodically act on classical (non-compact) spaces. However, none of these actions can be isometric in the sense of Goswami, leading to the conjecture that the result obtained by Goswami and Joardar about non-existence of genuine quantum isometry of classical compact connected Riemannian manifolds may hold in the non-compact case as well.

Paper Structure

This paper contains 11 sections, 10 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Locally compact quantum groups and their actions
  4. Bicrossed product of groups
  5. Existence of C*-actions of bicrossed products on spaces
  6. Properties of bicrossed product C*-actions
  7. Faithfulness
  8. Isometry
  9. Examples
  10. Baaj-Skandalis' ax+b group
  11. Split--Extension

Key Result

Theorem 3.1

There is a $\mathrm C^*$-action of $\mathbb G$ on $\textup{C}_0(\widehat{G_{1}})$.

Theorems & Definitions (27)

  • Definition 2.2: Kustermans-Vaes:LCQG*Definition 4.1
  • Definition 2.7
  • Definition 2.10
  • Definition 2.11: Vaes-Vainerman:Extension_of_lcqg*Definition 4.7
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.5
  • proof
  • Proposition 3.6
  • ...and 17 more