# when do norm-continuous unitary representations separate points of a group?

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?

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How important is the generality of all locally compact groups to you? If you restrict attention to 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
François, actually I'd like to have a theorem for general situation (not only for Lie groups, but a description for this subclass among all locally compact groups)... Is it possible that there is a gap between SIN-groups and those I am interested in?.. (Thank you for the reference anyway!) –  Sergei Akbarov Aug 11 '13 at 20:46
@FrancoisZiegler I think your last guess is correct (a theorem of Freudenthal and Weil IIRC) –  Yemon Choi Aug 12 '13 at 15:51