# $C_{∗}$-algebras associated with presentations of subshifts

### Kengo Matsumoto

Department of Mathematical Science Yokohama City University Sato 22-2, Kanazawa-ku, Yokohama 236-0027, Japan

## Abstract

A $λ$-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 $λ$-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 $λ$-graph systems among generating partial isometries and projections. If the $λ$-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.

## Cite this article

Kengo Matsumoto, $C_{∗}$-algebras associated with presentations of subshifts. Doc. Math. 7 (2002), pp. 1–30

DOI 10.4171/DM/115