TITLE: What does the representation theory of the reduced C*-algebra correspond to?
QUESTION [7 upvotes]: 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.)

REPLY [4 votes]: For suitable locally compact groups $G$ (separable, unimodular, type I), there is a measure $\mu$ on the dual $\hat{G}$ such that, for every function $f\in L^1(G)\cap L^2(G)$:
$$\int_G |f(g)|^2dg=\int_{\hat{G}}\|\pi(f)\|^2_{HS}d\mu(\pi)$$
where $\|.\|_{HS}$ denotes the Hilbert-Schmidt norm. The measure $\mu$ is the Plancherel measure of $G$ and its support is exactly the reduced dual, i.e. the dual of $C^*_r(G)$. For all this, see section 18.8 in J. Dixmier, $C^*$-algebras, North Holland, 1977.