$C^{\ast }$-algebras of labelled graphs III—$K$-theory computations. (6th October 2015)

• Record Type: Journal Article
• Authors: BATES, TERESA; CARLSEN, TOKE MEIER; PASK, DAVID
• Is Part Of: Ergodic theory and dynamical systems. Volume 37:Number 2(2017)
• Page Start: 337
• Page End: 368
• Publication Date: 2015-10-06
• Subjects: Ergodic theory -- Periodicals; Differentiable dynamical systems -- Periodicals; 515.42
• DOI: 10.1017/etds.2015.62
• Languages: English
• ISSNs: 0143-3857

Abstract

Abstract : 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 realizing 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 obtain 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 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 $C^{\ast }$ -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.