Abstract

In this paper we give a formula for the (Formula presented.)theory of the (Formula presented.)algebra of a weakly leftresolving labelled space. This is done by realizing the (Formula presented.)algebra of a weakly leftresolving labelled space as the Cuntz–Pimsner algebra of a (Formula presented.)correspondence. As a corollary, we obtain a gaugeinvariant uniqueness theorem for the (Formula presented.)algebra of any weakly leftresolving labelled space. In order to achieve this, we must modify the definition of the (Formula presented.)algebra of a weakly leftresolving 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.