A lambda-graph system for the Dyck shift and its $K$-groups

Abstract

A property of subshifts is described that allows to associate to the subshift a distinguishied presentation by a compact Shannon graph. For subshifts with this property and for the resulting invariantly associated compact Shannon graphs and their $\lambda$-graph systems the term $‘$Cantor horizon${}^{\prime }$ is proposed. The Dyck shifts are Cantor horizon. The $C^{\ast }$-algebras that are obtained from the Cantor horizon $\lambda$-graph systems of the Dyck shifts are separable, unital, nuclear, purely infinite and simple with UCT. The K-groups and Bowen-Franks groups of the Cantor horizon $\lambda$-graph systems of the Dyck shifts are computed and it is found that the $K_0$-groups are not finitely generated.