Cuntz-Pimsner Algebras of Group Representations

Given a locally compact group $G$ and a unitary representation $\rho:G\to U({\mathcal H})$onaHilbertspace${\mathcal H}$,weconstructa$C^$-correspondence${\mathcal E}(\rho)={\mathcal H}\otimes_{\mathbb C} C^(G)$over$C^(G)$andstudytheCuntz-Pimsneralgebra${\mathcal O}{{\mathcal E}(\rho)}$\mathrm{.}Weprovethatfor$G$compact,${\mathcal O}{{\mathcal E}(\rho)}$isstrongMoritaequivalenttoagraph$C^$-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 $C^{\ast }(G)$. 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 $C^{\ast }$-correspondence.