What does the representation theory of the reduced C*-algebra correspond to?

So there is a similar property.

Now $C^*_r(G)$ is the $C^star$-algebra generated by the left-regular rep. It a general theorem that if you have a unitary rep $pi:Grightarrow mathcal{U} (H)$, and if $rho: Grightarrow mathcal{U}(K)$ is another unitary rep that is weakly contained ($rhoprecpi$) in $pi$, then there is a surjective map from the reduced $C^star$-algebra to the algebra generated by $rho(G)subset B(K)$

So $C^star_r(G)$ surjects onto all reps that weakly contain the left-regular.

Note: $C^star_r(G)simeq C^star(G)$ iff G is amenable.

A good source for most of this

http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf

This is the pdf of a book about Property (T). Appendix F.4 is about the above questions but the whole book is of interest for people in operator algebras, representation theory, geometric group theory, and many other fields.

EDIT: Another good source, which is directed to Yemon's comment is http://arxiv.org/PS_cache/math/pdf/0509/0509450v1.pdf

This is a survey, by Pierre de la Harpe, of groups whose reduced $C^star$-algebra is simple.