Fell bundles over a countable discrete group and strong Morita equivalence for inclusions of $C^*$-algebras

TL;DR

This paper investigates the relationship between two saturated Fell bundles over a countable discrete group, showing that strong Morita equivalence of their associated inclusions implies the bundles are equivalent up to a group automorphism.

Contribution

It establishes a link between strong Morita equivalence of inclusions of $C^*$-algebras and the equivalence of the underlying Fell bundles under automorphisms of the group.

Findings

Strong Morita equivalence implies Fell bundle equivalence up to automorphism.

Irreducibility condition leads to trivial relative commutant.

Results connect bundle equivalence with algebraic automorphisms.

Abstract

We consider two saturated Fell bundles over a countable discrete group, whose unit fibers are $\sigma$-unital $C^*$-algebras. Then by taking the reduced cross-sectional $C^*$-algebras, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions of $C^*$-algebras is irreducible, that is, the relative commutant of one of the unit fiber algebras, which is a $\sigma$-unital $C^*$-algebra, in the multiplier $C^*$-algebra of the reduced cross-sectional $C^*$-algebra is trivial. We show that the two saturated Fell bundles are then equivalent up to some automorphism of the group.