Purely infinite simple reduced C*-algebras of one-relator separated graphs

TL;DR

This paper investigates the properties of reduced C*-algebras associated with separated graphs, establishing conditions under which they are purely infinite simple or have a faithful tracial state, extending known results to a broader class.

Contribution

It generalizes a result on purely infinite simple free products to the amalgamated case for separated graph C*-algebras, identifying when they are purely infinite simple.

Findings

Reduced C*-algebras are either purely infinite simple or have a faithful tracial state.

The main tool is a generalization of a free product result by Dykema.

Applicable to a large class of separated graphs.

Abstract

Given a separated graph $(E,C)$, there are two different C*-algebras associated to it, the full graph C*-algebra $C^*(E,C)$, and the reduced one $C^*_{\text{red}} (E,C)$. For a large class of separated graphs $(E,C)$, we prove that $C^*_{\text{red}} (E,C)$ either is purely infinite simple or admits a faithful tracial state. The main tool we use to show pure infiniteness of reduced graph C*-algebras is a generalization to the amalgamated case of a result on purely infinite simple free products due to Dykema.