Notes on C_0-representations and the Haagerup property

Abstract

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 $C^{\ast }(G)$ onto $C_{{\pi }_0}^{\ast }(G)$. 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.

Notes partly of an expository nature; 11 pages; a few typos fixed; final version to appear in the Bulletin of the Belgian Mathematical Society - Simon Stevin