The Zappa–Szép product of twisted groupoids

Abstract

We define and study the external and the internal Zappa–Szép 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–Szép twist $\Sigma$ over the Zappa–Szép product ${\mathcal{G}}_1\bowtie {\mathcal{G}}_2$. We prove that the resulting (reduced and full) twisted groupoid ${\mathrm{C}}^{\ast }$-algebra of the Zappa–Szép twist $\Sigma \to {\mathcal{G}}_1\bowtie {\mathcal{G}}_2$ is a ${\mathrm{C}}^{\ast }$-blend of its subalgebras corresponding to the subtwists ${\Sigma }_i\to {\mathcal{G}}_i$. Using Kumjian–Renault theory, we then prove a converse: Any ${\mathrm{C}}^{\ast }$-blend in which the intersection of the three algebras is a Cartan subalgebra in all of them, arises as the reduced twisted groupoid ${\mathrm{C}}^{\ast }$-algebras from such a Zappa–Szép twist $\Sigma \to {\mathcal{G}}_1\bowtie {\mathcal{G}}_2$ of two twists ${\Sigma }_1\to {\mathcal{G}}_1$ and ${\Sigma }_2\to {\mathcal{G}}_2$.