The Zappa-Sz\'{e}p product of twisted groupoids

Abstract

We define and study the external and the internal Zappa-Sz\'{e}p product of twists over groupoids. We determine when a pair $(\Sigma_{1},\Sigma_{2})$ of twists over a matched pair $(\mathcal{G}_{1},\mathcal{G}_{2})$ of groupoids gives rise to a Zappa-Sz\'{e}p twist $\Sigma$ over the Zappa-Sz\'{e}p product $\mathcal{G}_{1}\bowtie\mathcal{G}_{2}$. We prove that the resulting (reduced and full) twisted groupoid C*-algebra of the Zappa-Sz\'{e}p twist $\Sigma\to \mathcal{G}_{1}\bowtie\mathcal{G}_{2}$ is a C*-blend of its subalgebras corresponding to the subtwists $\Sigma_{i}\to \mathcal{G}_{i}$. Using Kumjian-Renault theory, we then prove a converse: Any C*-blend in which the intersection of the three algebras is a Cartan subalgebra in all of them, arises as the reduced twisted groupoid C*-algebras from such a Zappa-Sz\'{e}p twist $\Sigma\to \mathcal{G}_{1}\bowtie\mathcal{G}_{2}$ of two twists $\Sigma_{1}\to \mathcal{G}_{1}$ and $\Sigma_{2}\to \mathcal{G}_{2}$.