Pure infiniteness and paradoxicality for graph $C^*$-algebras

TL;DR

This paper establishes precise conditions under which the $C^*$-algebra of a row-finite graph without sinks is purely infinite, linking groupoid properties to algebraic infiniteness.

Contribution

It provides necessary and sufficient conditions for pure infiniteness of graph $C^*$-algebras based on groupoid properties, advancing understanding of their structure.

Findings

Pure infiniteness characterized by essential principal property.

Existence of paradoxical sets is both necessary and sufficient.

Conditions apply to row-finite graphs without sinks.

Abstract

We obtain necessary and sufficient conditions for pure infiniteness of the path groupoid $C^*$-algebra of a row-finite graph without sinks. In particular we show that for such a path groupoid $\mathcal{G}_E$, the properties of being essential principal and the existence of a basis of $(\mathcal{G}_E^a, 2, 1)$-paradoxical sets for the topology are not only sufficient, but also necessary.