$C^{\ast }$-algebras of labelled graphs III - $K$-theory computations

In this paper we give a formula for the $K$-theory of the $C^{\ast }$-algebra of a weakly left-resolving labelled space. This is done by realising the $C^{\ast }$-algebra of a weakly left-resolving labelled space as the Cuntz-Pimsner algebra of a $C^{\ast }$-correspondence. As a corollary we get a gauge invariant uniqueness theorem for the $C^{\ast }$-algebra of any weakly left-resolving labelled space. In order to achieve this we must modify the definition of the $C^{\ast }$-algebra of a weakly left-resolving labelled space. We also establish strong connections between the various classes of $C^{\ast }$-algebras which are associated with shift spaces and labelled graph algebras. Hence, by computing the $K$-theory of a labelled graph algebra we are providing a common framework for computing the $K$-theory of graph algebras, ultragraph algebras, Exel-Laca algebras, Matsumoto algebras and the $C^{\ast }$-algebras of Carlsen. We provide an inductive limit approach for computing the $K$-groups of an important class of labelled graph algebras, and give examples.