An index formula in a class of groupoid $C^*$-algebras

We consider the groupoid $C^*$-algebra $\mathcal{T} = C^*(\mathcal{G})$, where the groupoid $\mathcal{G}$ is a reduction of a transformation group $\mathcal{G} = (Y \times G)|X$, and $Y$

and $X$ are suitable topological spaces. We impose additional constraints on a cross-section $\psi$, which gives opportunity to define cyclic 1-cocycle and to obtain a formula that calculates the index of the Fredholm operators.