$C^{* }$-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.