THE BRAUER GROUP OF A LOCALLY COMPACT GROUPOID

We define the Brauer group Br (G) of a locally compact groupoid G to b e the set of Morita equivalence classes of pairs (A, alpha) consisting of an elementary C-bundle A over G((0)) satisfying Fell's condition and an action alpha of G on A by -isomorphisms. When G is the transfo rmation groupoid X 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 grou poids, then Br (G) and Br (H) are isomorphic. This generalizes the res ult 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 Br-0 (G) of Br (G) consisting of classes [A,alpha] with A having trivial Dixmier-Douady invariant is isomorphic to a quotient epsilon (G) of the collection Tw (G) of twists over G. Finally we prove that Br (G) is isomorphic to t he inductive limit Ext (G, T) of the groups epsilon (G(X)) where X var ies over all principal G spaces X and G(X) is the imprimitivity groupo id associated to X.