由拓扑马尔可夫链的Dyck系统产生的$C^*$ -代数
$C^*$-algebras arising from Dyck systems of topological Markov chains
摘要：
设$A$是一个元素在$\{0,1\}$中的$N \times N $不可约矩阵。本文定义了一个由$2N$括号$(_1,...,(_N,)_1,...,)_N$和标准括号规则以及来自$A$的Markov链规则组成的非sofic子移位$D_A$。子移位被认为是由Cuntz-Krieger代数${\Cal O}_A$的规范生成元$S_1^*,..., S_N^*, S_1,..., S_N $定义的子移位。我们构造了一个不可约的$λ$ -图系统${{\frak L}^{Ch(D_A)}}$，它表示子移位$D_A$，使得我们有一个相应的简单纯无限$C^*$ -代数${\Cal O}_{{\frak L}^{Ch(D_A)}}$ .我们证明了${\Cal O}_{{\frak L}^{Ch(D_A)}}$是一个泛唯一的$C^*$ -代数，它满足生成部分等距的$2N$之间的某些算子关系。给出了一些例子，使得它们不稳定同构于任何Cuntz-Krieger代数。
英文摘要：
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^*,..., S_N^*, S_1,..., S_N $ of the Cuntz-Krieger algebra ${\Cal O}_A$. We construct an irreducible $λ$-graph system ${{\frak L}^{Ch(D_A)}}$ that presents the subshift $D_A$ so that we have an associated simple purely infinite $C^*$-algebra ${\Cal O}_{{\frak L}^{Ch(D_A)}}$. We prove that ${\Cal O}_{{\frak L}^{Ch(D_A)}}$ 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.
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