On a construction due to Khoshkam and Skandalis

TL;DR

This paper studies the Wiener Hopf algebra associated with semigroup actions on C*-algebras, showing it can be represented as a groupoid crossed product and analyzing its K-theory in specific cases.

Contribution

It demonstrates that the Wiener Hopf algebra can be realized as a groupoid crossed product and establishes K-theory equivalence for free semigroup actions.

Findings

Wiener Hopf algebra is representable as a groupoid crossed product.

K-theory of the Wiener Hopf algebra coincides with that of the original algebra for free semigroups.

Provides a new perspective on the structure and invariants of Wiener Hopf algebras.

Abstract

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 $C^{*}$-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.