Completely bounded approximation property of semigroup crossed products

In this paper, we study the CBAP of the reduced semigroup crossed product $\mathcal{A}\rtimes_{\alpha,r} P$ for a quasi-lattice ordered group $(G,P)$. We prove that if $G$ is amenable and $\mathcal{A}$ has the CBAP, then $\mathcal{A}\rtimes_{\alpha,r} P$ has the CBAP with the same Haagerup constant. We relate the CBAP of $\mathcal{A}\rtimes_{\alpha,r} P$ with an approximation property for completely bounded kernels over $P$. When $P$ is right Ore, we introduce and characterise the weak amenability of $P$ as the CBAP of the reduced semigroup $C^*$-algebra with respect to the canonical tracial state and the linear space of finitely supported functions on $P\times P$.