Let $G$ be a locally compact group. The group C\*-algebra $C^\* (G)$ is designed to come with a natural bijection between its (nondegenerate) representations and the (strongly continuous, unitary) representations of $G$.


*Question*: Is there a similar statement for the reduced group C\*-algebra $C^\*\_r (G)$?


If the answer is no, I'll probably end up asking for the actual purpose of defining $C^\*\_r (G)$. So far, I know that its isomorphic to $C^\* (G)$ in important cases, and that its construction is in some sense simpler than the one of $C^\* (G)$.


(The definitions and the claims used above can be found in Blackadar's *Operator Algebras*.) 



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 $\pi:G\rightarrow \mathcal{U} (H)$, and if $\rho: G\rightarrow \mathcal{U}(K)$ is another unitary rep that is weakly contained ($\rho\prec\pi$) 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. 


