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 ε (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 ε(G^X) where X varies over all principal G spaces X and G^X is the imprimitivity groupoid associated to X.