On a construction due to Khoshkam and Skandalis

In this paper, we consider the Wiener Hopf algebra, denoted $\mathcal{W}(A,P,G,α)$, associated to an action of a discrete subsemigroup $P$ of a group $G$ on a $C^{*}$-algebra $A$. We show that $\mathcal{W}(A,P,G,α)$ 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 $\mathcal{W}(A,P,G,α)$ and the $K$-theory of $A$ coincides.

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