# Amenability and approximation properties for partial actions and Fell bundles

## TL;DR

This paper introduces AD-amenability for partial actions and Fell bundles, establishing its connection to nuclearity of cross-sectional C*-algebras and showing its preservation under equivalence and relation to approximation properties.

## Contribution

It defines AD-amenability for partial actions and Fell bundles, linking it to nuclearity and approximation properties, and proves key equivalence and preservation results.

## Key findings

- AD-amenability characterizes nuclearity of cross-sectional C*-algebras.
- AD-amenability is equivalent to an approximation property by Exel.
- AD-amenability is preserved under Fell bundle equivalence.

## Abstract

Building on previous papers by Anantharaman-Delaroche (AD) we introduce and study the notion of AD-amenability for partial actions and Fell bundles over discrete groups. We prove that the cross-sectional C*-algebra of a Fell bundle is nuclear if and only if the underlying unit fibre is nuclear and the Fell bundle is AD-amenable. If a partial action is globalisable, then it is AD-amenable if and only if its globalisation is AD-amenable. Moreover, we prove that AD-amenability is preserved by (weak) equivalence of Fell bundles and, using a very recent idea of Ozawa and Suzuki, we show that AD-amenabity is equivalent to an approximation property introduced by Exel.