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 BrH(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 BrG(X/H) and BrH (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(GX) where X varies over all principal G spaces X and GX is the imprimitivity groupoid associated to X.


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pp. 901-954
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