$K$-theory and homotopies of 2-cocycles on group bundles

Abstract

This paper continues the author's program of investigating the question of when a homotopy of 2-cocycles $\Omega = \{\omega _t\}_{t \in [0,1]}$ on a locally compact Hausdorff groupoid $\mathcal{G} $ induces an isomorphism of the $K$-theory groups of the twisted groupoid $C^*$-algebras: \[ K_*(C^*(\mathcal{G} , \omega _0)) \cong K_*(C^*(\mathcal{G} , \omega _1)). \] Building on our earlier work in \cite {eag-kgraph, transf-gps}, we show that, if $\pi : \mathcal{G} \to M$ is a locally trivial bundle of amenable groups over a locally compact Hausdorff space $M$, a homotopy $\Omega = \{\omega _t\}_{t \in [0,1]}$ of 2-cocycles on $\mathcal{G} $ gives rise to an isomorphism: \[ K_*(C^*(\mathcal{G} , \omega _0)) \cong K_*(C^*(\mathcal{G} , \omega _1)). \]