On simple labelled graph C*-algebras

We consider the simplicity of the C*-algebra associated to a labelled space (E, G, E), where (E, L) is a labelled graph and (epsilon) over bar is the smallest accommodating set containing all generalized vertices. We prove that if C*(E, L, (epsilon) over bar ) is simple, then (E, L, (epsilon) over bar) is strongly cofinal, and if, in addition, {v} is an element of (epsilon) over bar for every vertex v, then (E, L, (epsilon) over bar) is disagreeable. It is observed that C* (E, L, (epsilon) over bar) is simple whenever (E, L, (epsilon) over bar) is strongly cofinal and disagreeable, which is recently known for the C*-algebra C* (E, L, epsilon(0,-)) associated to a labelled space (E, L, epsilon(0,-)) of the smallest accommodating set epsilon(0,-). (C) 2011 Elsevier Inc. All rights reserved.