Ideal structure and pure infiniteness of ample groupoid $C^*$-algebras

TL;DR

This paper characterizes the ideal structure and pure infiniteness of reduced $C^*$-algebras of ample groupoids, linking algebraic properties to topological and paradoxical features of the groupoid's unit space.

Contribution

It provides new criteria for pure infiniteness and simplicity of groupoid $C^*$-algebras based on the structure of the underlying groupoid and introduces the type semigroup concept.

Findings

Characterization of ideal correspondence with open invariant subsets.

Conditions for pure infiniteness based on projections and paradoxicality.

Dichotomy result: simple $C^*$-algebras are either stably finite or purely infinite.

Abstract

In this paper, we study the ideal structure of reduced $C^*$-algebras $C^*_r(G)$ associated to \'etale 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.