Characterizations of amenability for noncommutative dynamical systems and Fell bundles

Abstract

We resolve key open questions regarding approximation properties and their permanence for Fell bundles over locally compact groups. Specifically, we establish the equivalence between the B\'edos--Conti approximation property (BCAP) and the Exel--Ng positive approximation property (AP), completely removing the necessity of assuming nuclearity on the unit fiber. To overcome the obstructions present in general Fell bundles (such as the lack of spatial arguments and exactness), we introduce a diagonal maximal tensor product $\otimes^d_{\max}$. We prove that a Fell bundle $\mathcal{A}$ has the AP if and only if $\mathcal{A} \otimes^d_{\max} \mathcal{B}$ has the weak containment property (wcp) for every Fell bundle $\mathcal{B}$. For $C^*$-dynamical systems, this yields a characterization of amenability that was known to hold under exactness assumptions. Furthermore, this tensorial machinery allows us to establish highly non-trivial permanence properties for the AP, including passage to restrictions over closed subgroups and partial quotients by normal subgroups. We also provide applications concerning the nuclearity of full and reduced cross-sectional $C^*$-algebras.