The Brauer group of a locally compact groupoid

We define the Brauer group Br ( G ) of a locally compact groupoid G to be the set of Morita equivalence classes of pairs ( A , α) consisting of an elementary C *-bundle A over G (0) satisfying Fell's condition and an action α of G on A by *-isomorphisms. When G is the transformation groupoid X × H , then Br( G ) is the equivariant Brauer group Br H ( X ). In addition to proving that Br( G ) is a group, we prove three isomorphism results. First we show that if G and H are equivalent groupoids, then Br( G ) and Br( H ) are isomorphic. This generalizes the result that if G and H are groups acting freely and properly on a space X , say G on the left and H on the right, then Br G ( X / H ) and Br H ( G \ X ) are isomorphic. Secondly we show that the subgroup Br0 ( G ) of Br ( G ) consisting of classes [ A , α] with A having trivial Dixmier-Douady invariant is isomorphic to a quotient e ( G ) of the collection Tw ( G ) of twists over G . Finally we prove that Br( G ) is isomorphic to the inductive limit Ext( G , [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]) of the groups e( G X ) where X varies over all principal G spaces X and G X is the imprimitivity groupoid associated to X .