  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.
