Labelled space $C^{\ast }$-algebras as partial crossed products and a simplicity characterization

A partial action is associated with a normal weakly left resolving labelled space such that the crossed product and labelled space $C^{\ast }$-algebras are isomorphic. An improved characterization of simplicity for labelled space $C^{\ast }$-algebras is given and applied to $C^{\ast }$-algebras of subshifts.