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 strong 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.