Generalised Twisted Groupoids and their C*-algebras

We consider a locally compact Hausdorff groupoid $G$, and twist by a more general locally compact Hausdorff abelian group $\mathrm{\Gamma }$ rather than the complex unit circle $\mathbb{T}$. We investigate the construction of $C^{\ast }$-algebras in analogue to the usual twisted groupoid $C^{\ast }$-algebras, and we show that, in fact, any $\mathrm{\Gamma }$-twisted groupoid $C^{\ast }$-algebra is isomorphic to a usual twisted groupoid $C^{\ast }$-algebra.