  A well-known theorem of Day and Dixmier states that any uniformly bounded
representation of an amenable locally compact group $G$ on a Hilbert space is
similar to a unitary representation. Within the category of locally compact
quantum groups, the conjectural analogue of the Day-Dixmier theorem is that
every completely bounded Hilbert space representation of the convolution
algebra of an amenable locally compact quantum group should be similar to a
$\ast$-representation. We prove that this conjecture is false for a large class
of non-Kac type compact quantum groups, including all $q$-deformations of
compact simply connected semisimple Lie groups. On the other hand, within the
Kac framework, we prove that the Day-Dixmier theorem does indeed hold for
several new classes of examples, including amenable discrete quantum groups of
Kac-type.
