Zappa–Szép product groupoids and C∗-blends

We study the external and internal Zappa–Szép product of topological groupoids. We show that under natural continuity assumptions the Zappa–Szép product groupoid is étale if and only if the individual groupoids are étale. In our main result we show that the C*-algebra of a locally compact Hausdorff étale Zappa–Szép product groupoid is a C*-blend, in the sense of Exel, of the individual groupoid C*-algebras. We finish with some examples, including groupoids built from C*-commuting endomorphisms, and skew product groupoids.