Graph inverse semigroups, groupoids and their C*-algebras

TL;DR

This paper develops a comprehensive theory of graph C*-algebras using path groupoids and inverse semigroups, extending to graphs with vertices emitting infinitely many edges, and proves amenability and simplicity criteria.

Contribution

It introduces a new framework for graph C*-algebras that does not require row finiteness, utilizing path groupoids and inverse semigroups.

Findings

Path groupoid is shown to be amenable.

Provides a groupoid-based proof of simplicity characterization.

Extends theory to graphs with infinite emitters.

Abstract

We develop a theory of graph C*-algebras using path groupoids and inverse semigroups. Row finiteness is not assumed so that the theory applies to graphs for which there are vertices emitting a countably infinite set of edges. We show that the path groupoid is amenable, and give a groupoid proof of a recent theorem of Szymanski characterizing when a graph C*-algebra is simple.