Weak containment and conjugate actions for Fell-bundle actions

Define weak containment and conjugate actions for actions of a Fell bundle on Hilbert bundles, if possible, and determine whether these notions support a corresponding theory of amenability.

Background

The paper’s tensor product construction motivates questions about representation-theoretic notions beyond those established in the article. In particular, the authors ask whether weak containment and conjugation can be formulated for actions of a Fell bundle on Hilbert bundles.

They further propose an analogue of amenability for such actions, modeled on Bekka’s notion for unitary group representations, by requiring the trivial action to be weakly contained in the tensor product of an action with its conjugate.

References

Can one define weak containment for $$-actions on Hilbert $$-bundles? And is there a meaningful way to define the conjugate of a $$-action? If so, can one then define a concept of amenability for a $$-action, similar to what Bekka did for unitary representations of groups , for instance by saying that the trivial $$-action is weakly contained in the tensor product of the $$-action with its conjugate?

— Actions of Fell bundles, II: tensor products and Fourier-Stieltjes algebras  (2610.13646 - Bédos et al., 9 Oct 2026) in Section 7, “Some questions” (Section \ref{final})