An equivalence theorem for reduced Fell bundle C*-algebras

We show that if E is an equivalence of upper semicontinu- ous Fell bundles B and C over groupoids, then there is a linking bundle L(E ) over the linking groupoid L such that the full cross-sectional alge- bra of L(E ) contains those of B and C as complementary full corners, and likewise for reduced cross-sectional algebras. We show how our re- sults generalise to groupoid crossed-products the fact, proved by Quigg and Spielberg, that Raeburn's symmetric imprimitivity theorem passes through the quotient map to reduced crossed products.