Group actions on graphs and $C^{\ast }$-correspondences

Abstract

If $G$ acts on a $C^{\ast }$-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 ${\mathcal{O}}_{\mathcal{H}}⋊G$ 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 $C^{\ast }$-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 $C^{\ast }(E)⋊G$ is isomorphic to the $C^{\ast }$-algebra of a graph of $C^{\ast }$-correspondences and stably isomorphic to a locally finite graph algebra. If $C^{\ast }(E)$ is simple and purely infinite and the action of $G$ is outer, then $C^{\ast }(E{)}^G$ and $C^{\ast }(E)⋊G$ are also simple and purely infinite with the same $K$-theory groups. We illustrate with several examples.

To appear Houston J. Math

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• Groupoid actions on $C^{\ast }$-correspondences