Zappa-Sz\'{e}p product of a Fell bundle and a groupoid

TL;DR

This paper introduces a new construction called the Zappa-Szép product for Fell bundles over groupoids, explores its properties, and relates it to covariant representations and universal C*-algebras, generalizing previous results.

Contribution

It defines the Zappa-Szép product of a Fell bundle and a groupoid, and shows how all such bundles can be obtained this way under certain conditions.

Findings

The Zappa-Szép product of a Fell bundle is itself a Fell bundle over the product of groupoids.

The associated universal C*-algebra is a C*-blend, extending earlier work on groupoid C*-algebras.

In the discrete case, the universal C*-algebra embeds injectively into the product's C*-algebra.

Abstract

We define the Zappa-Sz\'{e}p product of a Fell bundle by a groupoid, which turns out to be a Fell bundle over the Zappa-Sz\'{e}p product of the underlying groupoids. Under certain assumptions, every Fell bundle over the Zappa-Sz\'{e}p product of groupoids arises in this manner. We then study the representation associated with the Zappa-Sz\'{e}p product Fell bundle and show its relation to covariant representations. Finally, we study the associated universal C*-algebra, which turns out to be a C*-blend, generalizing an earlier result about the Zappa-Sz\'{e}p product of groupoid C*-algebras. In the case of discrete groups, the universal C*-algebra of a Fell bundle embeds injectively inside the universal C*-algebra of the Zappa-Sz\'{e}p product Fell bundle.