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

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.