Title: Ideal structure and pure infiniteness of ample groupoid $C^*$-algebras,
Abstract: In this paper, we study the ideal structure of reduced $C^*$-algebras
$C^*_r(G)$ associated to étale groupoids $G$. In particular, we characterize
when there is a one-to-one correspondence between the closed, two-sided ideals
in $C_r^*(G)$ and the open invariant subsets of the unit space $G^{(0)}$ of
$G$. As a consequence, we show that if $G$ is an inner exact, essentially
principal, ample groupoid, then $C_r^*(G)$ is (strongly) purely infinite if and
only if every non-zero projection in $C_0(G^{(0)})$ is properly infinite in
$C_r^*(G)$. We also establish a sufficient condition on the ample groupoid $G$
that ensures pure infiniteness of $C_r^*(G)$ in terms of paradoxicality of
compact open subsets of the unit space $G^{(0)}$.
Finally, we introduce the type semigroup for ample groupoids and also obtain
a dichotomy result: Let $G$ be an ample groupoid with compact unit space which
is minimal and topologically principal. If the type semigroup is almost
unperforated, then $C_r^*(G)$ is a simple $C^*$-algebra which is either stably
finite or strongly purely infinite.