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 C0(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.

We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions.

Amenability for groups can be extended to metric spaces, algebras over commutative fields and C * -algebras by adapting the notion of Følner nets. In the present article we investigate the close ties among these extensions and show that these three pictures unify in the context of the uniform Roe algebra C * u (X) over a metric space (X, d) with bounded geometry. In particular, we show that the following conditions are equivalent: (1) (X, d) is amenable;(2) the translation algebra generating C

In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite Communicated by Efim Zelmanov. extended metric spaces and for general algebras over fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of algebras: the unital Leavitt path algebras and the translation algebras on locally finite extended metric spaces. In particular, we show that the amenability of a metric space is equivalent to the algebraic amenability of the corresponding translation algebra.

We give some new characterizations of exactness for locally compact second countable groups. In particular, we prove that a locally compact second countable group is exact if and only if it admits a topologically amenable action on a compact Hausdorff space. This answers an open question by Anantharaman-Delaroche.