The Zappa–Szép product of twisted groupoids

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}^{*} -algebra of the Zappa–Szép twist \Sigma\to\mathcal{G}_{1}\bowtie\mathcal{G}_{2} is a \mathrm{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 \mathrm{C}^{*} -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}^{*} -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} .