The Brauer Group of a Locally Compact Groupoid

Details The Brauer Group of a Locally Compact Groupoid The Brauer Group of a Locally Compact Groupoid

1997-06-17

arxiv.org/abs/funct-an/9706004v1

Mathematics

Operator Algebras

52 pages AMS-LaTeX

Scientific paper

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 $\alpha$ of $G$ on $\A$ by $*$-isomorphisms. When $G$ is the transformation groupoid $X\times H$, 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 $X$.

Affiliated with

Also associated with

No associations

Rating

The Brauer Group of a Locally Compact Groupoid does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The Brauer Group of a Locally Compact Groupoid, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Brauer Group of a Locally Compact Groupoid will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-516759