The Brauer Group of a Locally Compact Groupoid
Alex Kumjian, Paul S. Muhly, Jean N. Renault, Dana P. Williams
Abstract
We define the Brauer group $\Br(G)$ of a locally compact groupoid
G
to be the set of Morita equivalence classes of pairs $(\A,\alpha)$ 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 $\Br_0(G)$ of $\Br(G)$ consisting of classes $[\A,\alpha]$ 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
.