On simple labelled graph $C^*$-algebras

Abstract

We consider the simplicity of the $C^*$-algebra associated to a labelled space $(E,\CL,\bE)$, where $(E,\CL)$ is a labelled graph and $\bE$ is the smallest accommodating set containing all generalized vertices. We prove that if $C^*(E, \CL, \bE)$ is simple, then $(E, \CL, \bE)$ is strongly cofinal, and if, in addition, $\{v\}\in \bE$ for every vertex $v$, then $(E, \CL, \bE)$ is disagreeable. It is observed that $C^*(E, \CL, \bE)$ is simple whenever $(E, \CL, \bE)$ is strongly cofinal and disagreeable, which is recently known for the $C^*$-algebra $C^*(E, \CL, \CEa)$ associated to a labelled space $(E, \CL, \CEa)$ of the smallest accommodating set $\CEa$.