Nuclearity and Exactness for Groupoid Crossed Products

Let $(\mathcal{A}, G, \alpha)$ be a groupoid dynamical system. We show that if $G$ is assumed to be measurewise amenable and the section algebra $A = \Gamma_0(G^{(0)}, \mathcal{A})$ is nuclear, then the associated groupoid crossed product is also nuclear. This generalizes an earlier result of Green for crossed products by locally compact groups. We also extend a related result of Kirchberg to groupoids. In particular, if $A$ is exact and $G$ is amenable, then we show that $\mathcal{A} \rtimes G$ is exact.