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

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 realising 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 get 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 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 $C^{\ast }$-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.

27 pages. We have made substantial changes in version 2 in order to avoid been affected by a mistake in the paper "$C^{\ast }$-algebras of labelled graphs I"

References in corpus (6)

• A unified approach to Exel-Laca algebras and C*-algebras associated to graphs
• Computing K-theory and Ext for graph C*-algebras
• C*-algebras of labelled graphs
• Cuntz-Pimsner C*-algebras associated with subshifts
• C*-crossed products and shift spaces
• Symbolic Dynamics, Partial Dynamical Systems, Boolean Algebras and C*-algebras Generated by Partial Isometries

Cited by in corpus (5)

• $C^{\ast }$-algebras associated to Boolean dynamical systems
• Gauge-invariant ideals of C*-algebras of Boolean dynamical systems
• AF labeled graph $C^{\ast }$-algebras
• The simplicity of the C*-algebras associated to arbitrary labeled spaces
• AF-embeddable labeled graph $C^{\ast }$-algebras