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

Abstract 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.