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$C^{\ast }$-algebras of labelled graphs III—$K$-theory computations

Published online by Cambridge University Press:  06 October 2015

TERESA BATES ,
TOKE MEIER CARLSEN  and
DAVID PASK
TERESA BATES
Affiliation:
School of Mathematics and Statistics, University of New South Wales, UNSW Sydney, NSW 2052, Australia email teresa@unsw.edu.au
TOKE MEIER CARLSEN
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway email Toke.Meier.Carlsen@math.ntnu.no
DAVID PASK
Affiliation:
School of Mathematics and Applied Statistics, Austin Keane Building (15), University of Wollongong, NSW 2522, Australia email dpask@uow.edu.au
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 2 , April 2017 , pp. 337 - 368
Copyright
© Cambridge University Press, 2015 

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