A Classic Morita Equivalence Result for Fell Bundle C∗-algebras

We show how to extend a classic Morita Equivalence Result of Green’s to the C∗-algebras of Fell bundles over transitive groupoids. Specifically, we show that if p : B → G is a saturated Fell bundle over a transitive groupoid G with stability group H = G(u) at u ∈ G(0), then C∗(G,B) is Morita equivalent to C∗(H,C ), where C = B|H . As an application, we show that if p : B → G is a Fell bundle over a group G and if there is a continuous G-equivariant map σ : Prim A → G/H , where A = B(e) is the C∗-algebra of B and H is a closed subgroup, then C∗(G,B) is Morita equivalent to C∗(H,C I ) where C I is a Fell bundle over H whose fibres are A/I–A/I -imprimitivity bimodules and I = ⋂{P : σ(P ) = eH }. Green’s result is a special case of our application to bundles over groups.