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*.)