K-theory and homotopies of 2-cocycles on transformation groups

This paper constitutes a first step in the author's program to investigate 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 reduced twisted groupoid $C^*$-algebras: $K_*(C^*_r(\mathcal{G}, \omega_0)) \cong K_*(C^*_r(\mathcal{G}, \omega_1)).$ Generalizing work of Echterhoff, L\"uck, Phillips, and Walters from 2010, we show that if $\mathcal{G} = G \ltimes X$ is a second countable locally compact transformation group, then whenever $G$ satisfies the Baum-Connes conjecture with coefficients, a homotopy $\omega = \{\omega_t\}_{t \in [0,1]}$ of 2-cocycles on $G \ltimes X$ gives rise to an isomorphism $K_*(C^*_r(G \ltimes X, \omega_0)) \cong K_*(C^*_r(G \ltimes X, \omega_1)).$