On simple labelled graph -algebras

We consider the simplicity of the C ⁎ -algebra associated to a labelled space ( E , L , E ¯ ) , where ( E , L ) is a labelled graph and E ¯ is the smallest accommodating set containing all generalized vertices. We prove that if C ⁎ ( E , L , E ¯ ) is simple, then ( E , L , E ¯ ) is strongly cofinal, and if, in addition, { v } ∈ E ¯ for every vertex v, then ( E , L , E ¯ ) is disagreeable. It is observed that C ⁎ ( E , L , E ¯ ) is simple whenever ( E , L , E ¯ ) is strongly cofinal and disagreeable, which is recently known for the C ⁎ -algebra C ⁎ ( E , L , E 0 , − ) associated to a labelled space ( E , L , E 0 , − ) of the smallest accommodating set E 0 , − .