Zappa-Sz\'ep product groupoids and C*-blends

TL;DR

This paper investigates the properties of Zappa-Szé product groupoids, demonstrating that their C*-algebras form C*-blends of the component groupoid C*-algebras under certain conditions, with applications to examples like endomorphisms and skew products.

Contribution

It establishes that the C*-algebra of an étale Zappa-Szé product groupoid is a C*-blend of the individual groupoid C*-algebras, extending the understanding of their structure.

Findings

Zappa-Szé product groupoids are étale if and only if the factors are étale.

The C*-algebra of a locally compact Hausdorff étale Zappa-Szé product is a C*-blend.

Examples include groupoids from *-commuting endomorphisms and skew product groupoids.

Abstract

We study the external and internal Zappa-Sz\'ep product of topological groupoids. We show that under natural continuity assumptions the Zappa-Sz\'ep product groupoid is \'etale if and only if the individual groupoids are \'etale. In our main result we show that the C*-algebra of a locally compact Hausdorff \'etale Zappa-Sz\'ep 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 *-commuting endomorphisms, and skew product groupoids.