Groupoid actions on $C^*$-correspondences

Abstract

Let the groupoid $G$ with unit space $G^0$ act via a representation $\rho$ on a $C^*$-correspondence ${\mathcal H}$ over the $C_0(G^0)$-algebra $A$. By the universal property, $G$ acts on the Cuntz-Pimsner algebra ${\mathcal O}_{\mathcal H}$ which becomes a $C_0(G^0)$-algebra. The action of $G$ commutes with the gauge action on ${\mathcal O}_{{\mathcal H}}$, therefore $G$ acts also on the core algebra ${\mathcal O}_{\mathcal H}^{\mathbb T}$. We study the crossed product ${\mathcal O}_{\mathcal H}\rtimes G$ and the fixed point algebra ${\mathcal O}_{\mathcal H}^G$ and obtain similar results as in \cite{D}, where $G$ was a group. Under certain conditions, we prove that ${\mathcal O}_{\mathcal H}\rtimes G\cong {\mathcal O}_{\mathcal H\rtimes G}$, where $\mathcal H\rtimes G$ is the crossed product $C^*$-correspondence and that ${\mathcal O}_{\mathcal H}^G\cong{\mathcal O}_\rho$, where ${\mathcal O}_\rho$ is the Doplicher-Roberts algebra defined using intertwiners. The motivation of this paper comes from groupoid actions on graphs. Suppose $G$ with compact isotropy acts on a discrete locally finite graph $E$ with no sources. Since $C^*(G)$ is strongly Morita equivalent to a commutative $C^*$-algebra, we prove that the crossed product $C^*(E)\rtimes G$ is stably isomorphic to a graph algebra. We illustrate with some examples.