On a construction due to Khoshkam and Skandalis

In this paper, we consider the Wiener Hopf algebra, denoted $\mathcal{W}(A,P,G,\alpha )$, associated to an action of a discrete subsemigroup $P$ of a group $G$ on a $C^{\ast }$-algebra $A$. We show that $\mathcal{W}(A,P,G,\alpha )$ can be represented as a groupoid crossed product. As an application, we show that when $P={\mathbb{F}}_n^{+}$, the free semigroup on $n$ generators, the $K$-theory of $\mathcal{W}(A,P,G,\alpha )$ and the $K$-theory of $A$ coincides.