Simple labeled graph $C^*$-algebras are associated to disagreeable labeled spaces

TL;DR

This paper establishes that for simple labeled graph $C^*$-algebras, the associated labeled space must be disagreeable, extending known conditions from graph $C^*$-algebras to labeled spaces.

Contribution

It proves the converse of previous results, showing that simplicity of the $C^*$-algebra implies the labeled space is disagreeable.

Findings

Simplicity of the $C^*$-algebra implies the labeled space is disagreeable.

Extends conditions for simplicity from graphs to labeled spaces.

Provides a characterization of simplicity in labeled graph $C^*$-algebras.

Abstract

By a labeled graph $C^*$-algebra we mean a $C^*$-algebra associated to a labeled space $(E,\mathcal L,\mathcal E)$ consisting of a labeled graph $(E,\mathcal L)$ and the smallest normal accommodating set $\mathcal E$ of vertex subsets. Every graph $C^*$-algebra $C^*(E)$ is a labeled graph $C^*$-algebra and it is well known that $C^*(E)$ is simple if and only if the graph $E$ is cofinal and satisfies Condition (L). Bates and Pask extend these conditions of graphs $E$ to labeled spaces, and show that if a set-finite and receiver set-finite labeled space $(E,\mathcal L, \mathcal E)$ is cofinal and disagreeable, then its $C^*$-algebra $C^*(E,\mathcal L, \mathcal E)$ is simple. In this paper, we show that the converse is also true.