The Brauer Group of a Locally Compact Groupoid

1997/06/17 by Alex Kumjian, Paul S. Muhly, Kumjian, Alex +5 · 1 citation
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #funct-an #math.OA

paper · pdf · doi:10.48550/arxiv.funct-an/9706004

52 pages AMS-LaTeX

arxiv created 1997/06/17 · arxiv updated 2009/11/30

Abstract

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,T) 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.

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