On simple labelled graph C⁎-algebras

We consider the simplicity of the C⁎-algebra associated to a labelled space , where (E,L) is a labelled graph and is the smallest accommodating set containing all generalized vertices. We prove that if is simple, then is strongly cofinal, and if, in addition, for every vertex v, then is disagreeable. It is observed that is simple whenever is strongly cofinal and disagreeable, which is recently known for the C⁎-algebra C⁎(E,L,E0,−) associated to a labelled space (E,L,E0,−) of the smallest accommodating set E0,−.