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 \cs-algebras of Fell bundles over transitive groupoids. Specifically, we show that if $p:\text{B}\to G$ is a saturated Fell bundle over a transitive groupoid $G$ with stability group $H=G(u)$ at $u\in \text{go}$, then $\text{cs}(G,\text{B})$ is Morita equivalent to $\text{cs}(H,\text{CC})$, where $\text{CC}=\text{B}\text{restr}H$. As an application, we show that if $p:\text{B}\to G$ is a Fell bundle over a group $G$ and if there is a continuous $G$-equivariant map $\sigma :\text{Prim}A\to G\mathrm{/}H$, where $A=B(e)$ is the \cs-algebra of $\text{B}$ and $H$ is a closed subgroup, then $\text{cs}(G,\text{B})$ is Morita equivalent to $\text{cs}(H,{\text{CC}}^I)$ where ${\text{CC}}^I$ is a Fell bundle over $H$ whose fibres are $A\mathrm{/}I\text{sme}A\mathrm{/}I$-\ib s and $I=\bigcap \{P:\sigma (P)=eH\}$. Green’s result is a special case of our application to bundles over groups.