The Brauer Group of a Locally Compact Groupoid
Alex Kumjian, Paul S. Muhly, Jean N. Renault, Dana P. Williams

TL;DR
This paper defines the Brauer group for locally compact groupoids, proves its properties, and establishes isomorphisms with related structures, extending classical results to a broader groupoid context.
Contribution
It introduces a new definition of the Brauer group for locally compact groupoids and proves key isomorphism theorems, generalizing known group results to groupoids.
Findings
$ ext{Br}(G)$ is a group for a locally compact groupoid $G$.
$ ext{Br}(G)$ is invariant under groupoid equivalence.
$ ext{Br}_0(G)$ is isomorphic to a quotient of the twist group $ ext{Tw}(G)$.
Abstract
We define the Brauer group of a locally compact groupoid to be the set of Morita equivalence classes of pairs consisting of an elementary C*-bundle over satisfying Fell's condition and an action of on by -isomorphisms. When is the transformation groupoid , then is the equivariant Brauer group . In addition to proving that is a group, we prove three isomorphism results. First we show that if and are equivalent groupoids, then and are isomorphic. This generalizes the result that if and are groups acting freely and properly on a space , say on the left and on the right then and are isomorphic. Secondly we show that the subgroup of consisting of classes with having trivial…
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology
