Coefficient realization of the identity bundle map in the nonunital case

Determine whether the identity bundle map on a nonunital Fell bundle can always be represented as a coefficient of an action on a Hilbert bundle over the unit fibre.

Background

For a unital Fell bundle, the paper shows that the Fourier–Stieltjes algebra consists of coefficient bundle maps of actions and observes that the identity bundle map is realized as a coefficient of the trivial action.

The authors note that the same coefficient realization is not established for nonunital Fell bundles. They provide an approximate-unit formula for the identity maps, but leave open whether an exact coefficient realization always exists.

References

If $$ is not unital, it is not clear %to us whether $I = (I_g)_{g\in G}$ can always be written as a coefficient of some $$-action (over the unit fibre).

— Actions of Fell bundles, II: tensor products and Fourier-Stieltjes algebras  (2610.13646 - Bédos et al., 9 Oct 2026) in Remark following Proposition \ref{FS-coeff} in Section \ref{FS}