Operator algebra seminar
Speaker: Alexander Kumjian (University of Nevada)
Title: Obstructions to lifting cocycles on groupoids and the associated $C^*$-algebras (joint work with Marius Ionescu)
Abstract: Let $\Gamma$ be an amenable locally compact groupoid and let $A$ be a closed subgroup of a locally compact abelian group $B$. Given a $B/A$-valued 1-cocycle $\phi$ on $\Gamma$, there is a central extension $\Sigma_\phi$ of $\Gamma$ by $A$ which is trivial iff $\phi$ lifts to a $B$-valued cocycle. We prove that $C^*(\Sigma_\phi)$ is isomorphic to the induced algebra of the natural action of $(B/A)^\hat$ on $C^*(\Gamma)$. If there is time we will also consider a simple class of examples arising from Cech 1-cocycles.