  For any locally compact group $G$, we show the existence and uniqueness up to
quasi-equivalence of a unitary $C_0$-representation $\pi_0$ of $G$ such that
all coefficient functions of $C_0$-representations of $G$ are coefficient
functions of $\pi_0$. The present work, strongly influenced by the work of N.
Brown and E. Guentner (which dealt exclusively with discrete groups), leads to
new characterizations of the Haagerup property: if $G$ is second countable,
then it has that property if and only if the representation $\pi_0$ induces a
*-isomorphism of $C^*(G)$ onto $C^*_{\pi_0}(G)$. When $G$ is discrete, we also
relate the Haagerup property to relative strong mixing properties of the group
von Neumann algebra $L(G)$ into finite von Neumann algebras.
