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 $C(X)$ 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.

12 pages

References in corpus (4)

• Double crossed products of locally compact quantum groups
• Quantum Group of Isometries in Classical and Noncommutative Geometry
• Faithful compact quantum group actions on connected compact metrizable spaces
• Invariant subsets under compact quantum group actions

Cited by in corpus (1)

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