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?

connected Lie groups, then Corollary 5 of I. M. Singer, Uniformly continuous representations of Lie groups (1952: ams.org/mathscinet-getitem?mr=49201) suggests that only the products (compact Lie group) $\times$ (vector group) will satisfy your condition. (These are, I guess, precisely the connected Lie SIN-groups?) – Francois Ziegler Aug 11 '13 at 20:14