C*-algebras arising from Dyck systems of topological Markov chains

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Let A be an N × N irreducible matrix with entries in {0,1}. We define the topological Markov Dyck shift Dₐ 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 S₁*,..., Sₙ*, S₁,..., Sₙ of the Cuntz-Krieger algebra Oₐ. We construct an irreducible λ-graph system LCh(Dₐ) that presents the subshift Dₐ so that we have an associated simple purely infinite C*-algebra O LCh(Dₐ). We prove that O LCh(Dₐ) is a universal unique C*-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.