C^* -algebras of labelled graphs III—K-theory computations. (6th October 2015) - BATES, TERESA; CARLSEN, TOKE MEIER; PASK, DAVID
Abstract : In this paper we give a formula for the K -theory of the C^* -algebra of a weakly left-resolving labelled space. This is done by realizing the C^* -algebra of a weakly left-resolving labelled space as the Cuntz–Pimsner algebra of a C^* -correspondence. As a corollary, we obtain a gauge-invariant uniqueness theorem for the C^* -algebra of any weakly left-resolving labelled space. In order to achieve this, we must modify the definition of the C^* -algebra of a weakly left-resolving labelled space. We also establish strong connections between the various classes of C^* -algebras that 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^* -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.