Simple labeled graph $C^{\ast }$-algebras are associated to disagreeable labeled spaces

By a labeled graph $C^{\ast }$-algebra we mean a $C^{\ast }$-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 $C^{\ast }$-algebra $C^{\ast }(E)$ is a labeled graph $C^{\ast }$-algebra and it is well known that $C^{\ast }(E)$ 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 $C^{\ast }$-algebra $C^{\ast }(E,\mathcal{L},\mathcal{E})$ is simple. In this paper, we show that the converse is also true.