Group actions on graphs and $C^*$-correspondences

Abstract

If $G$ acts on a $C^*$-correspondence ${\mathcal H}$, then by the universal property $G$ acts on the Cuntz-Pimsner algebra ${\mathcal O}_{\mathcal H}$ and we study the crossed product ${\mathcal O}_{\mathcal H}\rtimes G$ and the fixed point algebra ${\mathcal O}_{\mathcal H}^G$. Using intertwiners, we define the Doplicher-Roberts algebra ${\mathcal O}_\rho$ of a representation $\rho$ of a compact group $G$ on ${\mathcal H}$ and prove that ${\mathcal O}_{\mathcal H}^G$ is isomorphic to ${\mathcal O}_\rho$. When the action of $G$ commutes with the gauge action on ${\mathcal O}_{{\mathcal H}}$, then $G$ acts also on the core algebras ${\mathcal O}_{\mathcal H}^{\mathbb T}$, where $\mathbb T$ denotes the unit circle. We give applications for the action of a group $G$ on the $C^*$-correspondence ${\mathcal H}_E$ associated to a directed graph $E$. If $G$ is finite and $E$ is discrete and locally finite, we prove that the crossed product $C^*(E)\rtimes G$ is isomorphic to the $C^*$-algebra of a graph of $C^*$-correspondences and stably isomorphic to a locally finite graph algebra. If $C^*(E)$ is simple and purely infinite and the action of $G$ is outer, then $C^*(E)^G$ and $C^*(E)\rtimes G$ are also simple and purely infinite with the same $K$-theory groups. We illustrate with several examples.