Fell absorption principles for tensor products of Fell-bundle actions

Determine whether tensoring an $(A,\varphi)$-action on a Hilbert bundle with the trivial or regular action yields a Fell absorption principle, including for tensor products with the trivial action on either side and with the regular action on either side.

Background

The paper defines trivial and regular actions associated with actions of Fell bundles and develops tensor product constructions that combine actions on Hilbert bundles. In the concluding questions, it asks whether these constructions exhibit an analogue of the Fell absorption phenomenon familiar from unitary representation theory.

The unresolved issue includes tensoring an (A,φ)(A,\varphi)-action with the trivial action on either side, as well as tensoring it with the regular action on the left or right.

References

What can be said about \rho_ \dot\boxtimes \rho, or \rho \dot\boxtimes \rho_? What about tensoring \rho on the left with the regular action \widecheck{\rho_}, or on the right with \widecheck{\rho_} ? (See for the definition of regular actions.) Do we get some kind of Fell absorption principle(s)?

— 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})