Linear cross-sections and Fredholm operators in a class groupoid $C^*$-algebras

We consider the groupoid $C^*$-algebra $\mathcal{T} = C^*(\mathcal{G})$, 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.