## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Notes on $C_0$-representations and the Haagerup property

Paul Jolissaint

#### 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 the coefficient functions of $C_0$-representations of $G$ are exactly the coefficient functions of $\pi_0$. The present work, strongly influenced by [4] (which dealt exclusively with discrete groups), leads to new characterizations of the Haagerup property: $G$ 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 and countable, we also relate the Haagerup property to relative strong mixing properties in the sense of [9] of the group von Neumann algebra $L(G)$ into finite von Neumann algebras.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 21, Number 2 (2014), 263-274.

Dates
First available in Project Euclid: 20 May 2014

https://projecteuclid.org/euclid.bbms/1400592624

Digital Object Identifier
doi:10.36045/bbms/1400592624

Mathematical Reviews number (MathSciNet)
MR3211015

Zentralblatt MATH identifier
1296.22006

#### Citation

Jolissaint, Paul. Notes on $C_0$-representations and the Haagerup property. Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 2, 263--274. doi:10.36045/bbms/1400592624. https://projecteuclid.org/euclid.bbms/1400592624