Purely infinite labeled graph $C^*$-algebras

TL;DR

This paper investigates the pure infiniteness of generalized Cuntz-Krieger algebras from labeled spaces, establishing conditions under which these algebras and their quotients are purely infinite, extending known results to a broader class.

Contribution

It characterizes pure infiniteness for labeled graph $C^*$-algebras using conditions like disagreeability, loops, and property (K), and relates it to proper infiniteness of generating projections.

Findings

Pure infiniteness (IH) holds if the labeled space is disagreeable and every vertex connects to a loop.

Under property (K)-like conditions, the algebra is purely infinite iff all generating projections are properly infinite.

All quotients of the algebra have property (IH) if and only if the algebra is purely infinite in the Kirchberg-R{ exto}rdam sense.

Abstract

In this paper, we consider pure infiniteness of generalized Cuntz-Krieger algebras associated to labeled spaces $(E,\mathcal{L},\mathcal{E})$. It is shown that a $C^*$-algebra $C^*(E,\mathcal{L},\mathcal{E})$ is purely infinite in the sense that every nonzero hereditary subalgebra contains an infinite projection (we call this property (IH)) if $(E, \mathcal{L},\mathcal{E})$ is disagreeable and every vertex connects to a loop. We also prove that under the condition analogous to (K) for usual graphs, $C^*(E,\mathcal{L},\mathcal{E})=C^*(p_A, s_a)$ is purely infinite in the sense of Kirchberg and R{\o}rdam if and only if every generating projection $p_A$, $A\in \mathcal{E}$, is properly infinite, and also if and only if every quotient of $C^*(E,\mathcal{L},\mathcal{E})$ has the property (IH).