On Nuclearity of $C^*$-algebras of Fell bundles over \'etale groupoids

TL;DR

This paper proves that the nuclearity of fibers in a Fell bundle over an amenable étale groupoid implies the nuclearity of its associated $C^*$-algebra, using tensor product constructions.

Contribution

It introduces minimal tensor products of Fell bundles with fixed $C^*$-algebras to establish nuclearity transfer.

Findings

Nuclear fibers imply nuclear $C^*$-algebra for the Fell bundle

Introduces tensor product framework for Fell bundles

Establishes nuclearity preservation under certain conditions

Abstract

In this paper, we show that if E is a Fell bundle over an amenable \'etale locally compact Hausdorff groupoid such that every fiber on the unit space is nuclear, then $C?^*r(E)$ is also nuclear. In order to show this result, we introduce (minimal) tensor products of Fell bundles with fixed $C^*$-algebras.