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 -graph system. It yields a Renault's groupoid -algebra by following Deaconu's construction. The class of these -algebras generalize the class of -algebras associated with subshifts and hence the class of Cuntz-Krieger algebras. They are unital, nuclear, unique -algebras subject to operator relations encoded in the structure of the -graph systems among generating partial isometries and projections. If the -graph systems are irreducible (resp. aperiodic), they are simple (resp. simple and purely infinite). K-theory formulae of these -algebras are presented so that we know an example of a simple and purely infinite -algebra in the class of these -algebras that is not stably isomorphic to any Cuntz-Krieger algebra.
Cite this article
Kengo Matsumoto, -algebras associated with presentations of subshifts. Doc. Math. 7 (2002), pp. 1–30DOI 10.4171/DM/115