Linear cross-sections and Fredholm operators in a class groupoid C∗-algebras

Abstract

We consider the groupoid $C^{\ast }$-algebra $\mathcal{T}=C^{\ast }(\mathcal{G})$, where the groupoid $\mathcal{G}$ is a Wiener-Hopf groupoid, i. e., $\mathcal{G}$ a reduction of a transformation group $\mathcal{G}=(Y\times G)|X$, and $Y$ and $X$ are suitable topological spaces. We give a method to construct continuous linear cross-sections using contractions in ${\mathcal{G}}^0$ – the unit space of $\mathcal{G}$.

We establish a criterion for an operator $T\in \mathcal{B}$ to be Fredholm.