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

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

2012-03-14

arxiv.org/abs/1203.3072v1

Mathematics

Operator Algebras

25 pages

Scientific paper

In this paper we give a formula for the $K$-theory of a labelled graph algebra when the labelled graph in question is left-resolving. 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.

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