$C^*$-algebras of labelled graphs III - $K$-theory computations
Abstract
In this paper we give a formula for the -theory of the -algebra of a weakly left-resolving labelled space. This is done by realising the -algebra of a weakly left-resolving labelled space as the Cuntz-Pimsner algebra of a -correspondence. As a corollary we get a gauge invariant uniqueness theorem for the -algebra of any weakly left-resolving labelled space. In order to achieve this we must modify the definition of the -algebra of a weakly left-resolving labelled space. We also establish strong connections between the various classes of -algebras which are associated with shift spaces and labelled graph algebras. Hence, by computing the -theory of a labelled graph algebra we are providing a common framework for computing the -theory of graph algebras, ultragraph algebras, Exel-Laca algebras, Matsumoto algebras and the -algebras of Carlsen. We provide an inductive limit approach for computing the -groups of an important class of labelled graph algebras, and give examples.
Keywords
Cite
@article{arxiv.1203.3072,
title = {$C^*$-algebras of labelled graphs III - $K$-theory computations},
author = {Teresa Bates and Toke Meier Carlsen and David Pask},
journal= {arXiv preprint arXiv:1203.3072},
year = {2017}
}
Comments
27 pages. We have made substantial changes in version 2 in order to avoid been affected by a mistake in the paper "$C^*$-algebras of labelled graphs I"