2009/12/06 by Ionescu, Marius, Williams, Dana P.
#46L05 #46L55 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.0912.1124
We show that if p:\B→ G is a Fell bundle over a locally compact groupoid G and that A=Γ0(G(0);\B) is the \cs-algebra sitting over G(0), then there is a continuous G-action on \Prim A that reduces to the usual action when \B comes from a dynamical system. As an application, we show that if I is a G-invariant ideal in A, then there is a short exact sequence of \cs-algebras \xymatrix0\ar[r]&\cs(G,\BI)\ar[r] &\cs(G,\B)\ar[r]&\cs(G,\BqI)\ar[r]&0, where \cs(G,\B) is the Fell bundle \cs-algebra and \BI and \BqI are naturally defined Fell bundles corresponding to I and A/I, respectively. Of course this exact sequence reduces to the usual one for \cs-dynamical systems.