$C^{\ast }$-algebras arising from Dyck systems of topological Markov chains

Abstract

Let $A$ be an $N\times N$ irreducible matrix with entries in $\{0,1\}$. We define the topological Markov Dyck shift $D_A$ to be a nonsofic subshift consisting of the $2N$ brackets ${(}_1,...,{(}_N,{)}_1,...,{)}_N$ with both standard bracket rule and Markov chain rule coming from $A$. The subshift is regarded as a subshift defined by the canonical generators $S_1^{\ast },...,S_N^{\ast },S_1,...,S_N$ of the Cuntz-Krieger algebra ${\Cal O}_A$. We construct an irreducible $\lambda$-graph system ${\mathfrak{L}}^{Ch(D_A)}$ that presents the subshift $D_A$ so that we have an associated simple purely infinite $C^{\ast }$-algebra ${\Cal O}_{{\frak L}^{Ch(D_A)}}$. We prove that ${\Cal O}_{{\frak L}^{Ch(D_A)}}$ is a universal unique $C^{\ast }$-algebra subject to some operator relations among $2N$ generating partial isometries. Some examples are presented such that they are not stably isomorphic to any Cuntz-Krieger algebra.

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