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

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.