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

Valentin Deaconu  (University of Nevada, Reno)

14:00 pm to 15:00 am, May 20th, 2014   Science Building A1510

Abstract:

We consider actions of a group $G$ on directed graphs $E$ and on their $C^*$-algebras. When the action is free and proper, the crossed product $C^*(E)\rtimes_r G$ is strongly Morita equivalent to $C^*(E/G)$. For example, the fundamental group $\pi_1(E)$ acts freely and properly on the universal covering graph $\tilde{E}$ and $C^*(\tilde E)\rtimes_r\pi_1(E)$ is strongly Morita equivalent to $C^*(E)$.

For general actions, we define the Doplicher-Roberts algebra ${\mathcal O}_\rho$ of a representation $\rho$ of a compact group $G$ on the $C^*$-correspondence ${\mathcal H}_E$ and prove that ${\mathcal O}_\rho\cong C^*(E)^G$, the fixed point algebra. When the action of $G$ commutes with the gauge action on $C^*(E)$, we get examples of actions on the core algebras $C^*(E)^{\mathbb T}$.

If $G$ is finite and acts on a discrete graph $E$, we prove that $C^*(E)\rtimes G$ is isomorphic to the $C^*$-algebra of a graph of $C^*$-correspondences and stably isomorphic to a graph algebra. If $C^*(E)$ is simple and purely infinite and the action is outer, we prove that $C^*(E)^G$ and $C^*(E)\rtimes G$ are also simple and purely infinite with the same $K$-theory groups. We illustrate with several examples.