$C^*$-algebras of labelled graphs III - $K$-theory computations

TL;DR

This paper develops a formula for the K-theory of C*-algebras from weakly left-resolving labelled spaces, unifying various classes of shift space related C*-algebras and providing new computational methods.

Contribution

It introduces a K-theory formula for labelled space C*-algebras via Cuntz-Pimsner algebras, and connects different classes of shift space C*-algebras.

Findings

Established a gauge invariant uniqueness theorem.

Unified framework for K-theory computations across multiple C*-algebra classes.

Provided an inductive limit method for K-theory calculations.

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 realising the $C^*$-algebra of a weakly left-resolving labelled space as the Cuntz-Pimsner algebra of a $C^*$-correspondence. As a corollary we get 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 which 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.