The relative Haagerup property of semigroup crossed products

Abstract

In this paper, we study the relative Haagerup property of semigroup crossed products. Let $G$ be a lattice ordered group that acts on a unital C$^*$-algebra $\A$ through an action $\alpha$. We show that the inclusion $\A\subseteq \A\rtimes_{\alpha,r}G^+$ has the relative Haagerup property if and only if $G$ has the Haagerup property.