Generalised Twisted Groupoids and their C*-algebras
Lisa Orloff Clark, Michael Ó Ceallaigh, Hung Pham
TL;DR
The paper addresses whether generalised $\Gamma$-twists of a groupoid $G$ yield genuinely new $C^*$-algebras or can be realized as ordinary $\,\mathbb{T}$-twists. It develops a $\Gamma$-equivariant $\mathcal{C}_c$-algebra $\mathcal{C}_c(\Sigma; G; \chi)$ using a fixed character $\chi$ on $\Gamma$, constructs a Haar system on the twisted groupoid, and defines full and reduced $C^*$-algebras. The main result shows that for any character $\chi$, quotienting by $\ker\chi$ yields a $\Gamma/\ker\chi$-twist and a $*$-isomorphism between the corresponding algebras, reducing to a standard twisted $C^*$-algebra when the quotient is $\mathbb{T}$. If $\operatorname{im}(\chi)$ is finite, the algebras can be realized via a finite subgroup of $\mathbb{T}$. Overall, the findings imply that generalized twists do not produce new $C^*$-algebras beyond those from ordinary twists; the groupoid structure primarily governs the resulting theory.
Abstract
We consider a locally compact Hausdorff groupoid $G$, and twist by a more general locally compact Hausdorff abelian group $Γ$ rather than the complex unit circle $\mathbb{T}$. We investigate the construction of $C^*$-algebras in analogue to the usual twisted groupoid $C^*$-algebras, and we show that, in fact, any $Γ$-twisted groupoid $C^*$-algebra is isomorphic to a usual twisted groupoid $C^*$-algebra.