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

2009/12/06 by Ionescu, Marius, Williams, Dana P.
#46L55 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.0912.1125

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→ G is a saturated Fell bundle over a transitive groupoid G with stability group H=G(u) at u∈ \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→ 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 \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\setP:σ(P)=eH. Green's result is a special case of our application to bundles over groups.

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