DOCUMENTA MATHEMATICA, Vol. 7 (2002), 1-30

Kengo Matsumoto

$C^*$-Algebras Associated with Presentations of Subshifts

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 $C^*$-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 $\lambda$-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 $C^*$-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.

2000 Mathematics Subject Classification: Primary 46L35, Secondary 37B10.

Keywords and Phrases: $C^*$-algebras, subshifts, groupoids, Cuntz-Krieger algebras

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