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

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By a labeled graph C*-algebra we mean a C*-algebra associated to a labeled space (E,L,E) consisting of a labeled graph (E,L) and the smallest normal accommodating set 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,L, E) is cofinal and disagreeable, then its C*-algebra C*(E,L, E) is simple. In this paper, we show that the converse is also true.