  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.
