On the Haagerup Property of $C^*$-Dynamical Systems

Abstract:

      Let $A$ be a unital $C^*$-algebra with a state $\tau$ and $G$ be a discrete group that acts on $A$ through a $\tau$-preserving action $\alpha$. We first generalize the Haagerup property of dynamical systems by considering states and prove that the dynamical system has the Haagerup property if and only if the reduced crossed product does. Then we introduce the quasi-amenable action of $G$ on $A$ with respect to $\tau$. Finally, using the above results, we prove that if $\alpha$ is a quasi-amenable action of $G$ on $A$ with respect to $\tau$, then $(A,\tau)$ has the Haagerup property if and only if $(A\rtimes_{\alpha,r}G,\tau')$ does, where $\tau'$ is the induced state on $A\rtimes_{\alpha,r}G$. As a consequence, our main results improve some well known results in the classical situation.