Recently I found in the web a discussion on the following question:
For which locally compact group G its norm-contunuous unitary representations separate points of G?
(A unitary representation $\pi:G\to B(H)$ is said to be norm-continuous if it is continuous with respect to the usual norm topology on $B(H)$.)
As far as I understand, this happens quite rarely, however, there are some general results, in particular, Julia Kuznetsova writes in one of her papers, that every SIN-group $G$ has this property (see also a discussion in MO).
I wonder if there is a criterion? Is the class of locally compact groups $G$ satisfying this condition described?