Faithful actions of locally compact quantum groups on classical spaces

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.