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

Abstract

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