C∗-algebras of labelled graphs III—K-theory computations

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