$C^*$-algebras associated with presentations of subshifts

Abstract

A A-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 A-graph system. It yields a Renault's groupoid C*-algebra by following Deaconu's construction. The class of these C*-algebras generalize the class of C*-algebras associated with subshifts and hence the class of Cuntz-Krieger algebras. They are unital, nuclear, unique C*-algebras subject to operator relations encoded in the structure of the A-graph systems among generating partial isometries and projections. If the A-graph systems are irreducible (resp. aperiodic), they are simple (resp. simple and purely infinite). K-theory formulae of these C*-algebras are presented so that we know an example of a simple and purely infinite C*-algebra in the class of these C*-algebras that is not stably isomorphic to any Cuntz-Krieger algebra.