On the Haagerup property for partial crossed products

Abstract

Let $(A,G,\alpha)$ be a partial dynamical system and let $A\rtimes_{\alpha,r} G$ denote the associated reduced partial crossed product. In this article, we introduce the Haagerup property for partial actions of discrete groups on $C^*$-algebras. We prove that the partial crossed product $A\rtimes_{\alpha,r} G$ has the Haagerup property if and only if both $A$ and the partial action $\alpha$ have the Haagerup property. As a consequence, we obtain an equivalence between the Haagerup property of the partial crossed product and that of the underlying $C^*$-algebra and the acting group. We also show that the Haagerup property is preserved under inductive limits and apply this result to study the Haagerup property of inductive limits of partial crossed products.