Groupoid Dynamical Systems and Crossed Product, II — The Case of $C^{\ast }$-Systems

Abstract

By analogy with $C^{\ast }$-dynamical system, we define a $C^{\ast }$-groupoid dynamical system $(A,\Gamma ,\rho )$ where $A$ is a $C^{\ast }$-algebra, $\Gamma$ is a locally compact groupoid, and $\rho \to \mathrm{Aut}(A)$ is a continuous groupoid homomorphism. The groupoid crossed product $A{\times }_{\rho }\Gamma$ is defined and is shown to have similar properties as the case of a group action. As a special case of this situation, if $\rho$ is a continuous homomorphism from $\Gamma$ to a locally compact group $G$, we obtain groupoid dynamical system $(C_0(G),\Gamma ,\rho )$. In this case, there exists a co-action $\hat{\rho }$ of $G$ on $C^{\ast }(\Gamma )$ and the groupoid crossed product $C_0(G)\times \Gamma$ is isomorphic to the co-crossed product $C^{\ast }(\Gamma ){\ast }_{\hat{\rho }}G$ of $C^{\ast }(\Gamma )$ by $G$. The results in this paper is obtained by the analogy with our previous results for the case of $W^{\ast }$-systems.