Cuntz-Pimsner algebras of group representations

Abstract

Given a locally compact group $G$ and a unitary representation $\rho :G\to U({\mathcal H})$ on a Hilbert space $\mathcal {H}$, we construct a $C^*$-correspondence ${\mathcal E}(\rho )={\mathcal H}\otimes _{\mathbb C} C^*(G)$ over $C^*(G)$ and study the Cuntz-Pimsner algebra ${\mathcal O}_{{\mathcal E}(\rho )}$. We prove that, for $G$ compact, ${\mathcal O}_{{\mathcal E}(\rho )}$ is strongly Morita equivalent to a graph $C^*$-algebra. If $\lambda $ is the left regular representation of an infinite, discrete and amenable group $G$, we show that ${\mathcal O}_{{\mathcal E}(\lambda )}$ is simple and purely infinite, with the same $K$-theory as $C^*(G)$. If $G$ is compact abelian, any representation decomposes into characters and determines a skew product graph. We illustrate with several examples, and we compare ${\mathcal E}(\rho )$ with the crossed product $C^*$-correspondence.