The ideal structure of some analytic crossed products

We study the ideal structure of a class of some analytic crossed products. For an $r$-discrete, principal, minimal groupoid $G$, we consider the analytic crossed product $C^*(G,\sigma)\times _\alpha \mathbb{Z}_+$, where $\alpha$ is given by a cocycle $c$. We show that the maximal ideal space $\mathcal{M}$ of $C^*(G,\sigma)\times _\alpha \mathbb{Z}_+$ depends on the asymptotic range of $c$, $R_\infty(c)$; that is, $\mathcal{M}$ is homeomorphic to $\overline{\mathbb{D}}\mid R_\infty(c)$ for $R_\infty(c)$ finite, and $\cal M$ consists of the unique maximal ideal for $R_\infty(c)=\mathbb{T}$. We also prove that $C^*(G,\sigma)\times _\alpha \mathbb{Z}_+$ is semisimple in both cases, and that $R_\infty(c)$ is invariant under isometric isomorphism.