$C^{\ast }$-algebras associated with presentations of subshifts

Abstract

A $\lambda$-graph system is a labeled Bratteli diagram with an upward shift except the top vertices. We construct a continuous graph in the sense of V. Deaconu from a $\lambda$-graph system. It yields a Renault's groupoid $C^{\ast }$-algebra by following Deaconu's construction. The class of these $C^{\ast }$-algebras generalize the class of $C^{\ast }$-algebras associated with subshifts and hence the class of Cuntz-Krieger algebras. They are unital, nuclear, unique $C^{\ast }$-algebras subject to operator relations encoded in the structure of the $\lambda$-graph systems among generating partial isometries and projections. If the $\lambda$-graph systems are irreducible (resp. aperiodic), they are simple (resp. simple and purely infinite). K-theory formulae of these $C^{\ast }$-algebras are presented so that we know an example of a simple and purely infinite $C^{\ast }$-algebra in the class of these $C^{\ast }$-algebras that is not stably isomorphic to any Cuntz-Krieger algebra.