We introduce a notion of λ-graph bisystem that consists of a pair $({\mathfrak{L}}^{-},{\mathfrak{L}}^{+})$ of two labeled Bratteli diagrams ${\mathfrak{L}}^{-},{\mathfrak{L}}^{+}$ satisfying certain compatibility condition for labeling their edges. It is a two-sided extension of λ-graph system, that has been previously introduced by the author. Its matrix presentation is called a symbolic matrix bisystem. We first show that any λ-graph bisystem presents subshifts and conversely any subshift is presented by a λ-graph bisystem, called the canonical λ-graph bisystem for the subshift. We introduce an algebraically defined relation on symbolic matrix bisystems called properly strong shift equivalence and show that two subshifts are topologically conjugate if and only if their canonical symbolic matrix bisystems are properly strong shift equivalent. A λ-graph bisystem $({\mathfrak{L}}^{-},{\mathfrak{L}}^{+})$ yields a pair of $C^{⁎}$-algebras written ${\mathcal{O}}_{{\mathfrak{L}}^{-}}^{+},{\mathcal{O}}_{{\mathfrak{L}}^{+}}^{-}$ that are first defined as the $C^{⁎}$-algebras of certain étale groupoids constructed from $({\mathfrak{L}}^{-},{\mathfrak{L}}^{+})$. We study structure of the $C^{⁎}$-algebras, and show that they are universal unital unique $C^{⁎}$-algebras subject to certain operator relations among canonical generators of partial isometries and projections encoded by the structure of the λ-graph bisystem $({\mathfrak{L}}^{-},{\mathfrak{L}}^{+})$. If a λ-graph bisystem comes from a λ-graph system of a finite directed graph, then the associated subshift is the two-sided topological Markov shift $({\mathrm{\Lambda }}_A,{\sigma }_A)$ by its transition matrix A of the graph, and the associated $C^{⁎}$-algebra ${\mathcal{O}}_{{\mathfrak{L}}^{-}}^{+}$ is isomorphic to the Cuntz–Krieger algebra ${\mathcal{O}}_A$, whereas the other $C^{⁎}$-algebra ${\mathcal{O}}_{{\mathfrak{L}}^{+}}^{-}$ is isomorphic to the crossed product $C^{⁎}$-algebra $C({\mathrm{\Lambda }}_A){⋊}_{{\sigma }_A^{⁎}}\mathbb{Z}$ of the commutative $C^{⁎}$-algebra $C({\mathrm{\Lambda }}_A)$ of continuous functions on the shift space ${\mathrm{\Lambda }}_A$ of the two-sided topological Markov shift by the automorphism ${\sigma }_A^{⁎}$ induced by the homeomorphism of the shift ${\sigma }_A$. This phenomenon shows a duality between Cuntz–Krieger algebra ${\mathcal{O}}_A$ and the crossed product $C^{⁎}$-algebra $C({\mathrm{\Lambda }}_A){⋊}_{{\sigma }_A^{⁎}}\mathbb{Z}$.

我们引入由一对组成的λ-图双系统概念$（{\mathfrak{大号}}^{-}，{\mathfrak{大号}}^{+}）$ 标记的两个Bratteli图 ${\mathfrak{大号}}^{-}，{\mathfrak{大号}}^{+}$满足某些兼容性条件以标记其边缘。这是作者先前介绍的λ谱图系统的双面扩展。其矩阵表示称为符号矩阵双系统。我们首先表明，任何λ图双系统都存在子移位，反之，任何子移都由λ图双系统提出，该子移称为规范λ图双系统。我们在符号矩阵双系统上引入了代数定义的关系，称为适当的强移位等价关系，并表明，当且仅当两个子移位在其规范的符号矩阵双系统都具有适当的强移位等价关系时才是拓扑共轭的。一λ -图bisystem$（{\mathfrak{大号}}^{-}，{\mathfrak{大号}}^{+}）$ 产生一对 $C^{⁎}$-代数写 ${\mathcal{Ø}}_{{\mathfrak{大号}}^{-}}^{+}，{\mathcal{Ø}}_{{\mathfrak{大号}}^{+}}^{-}$ 首先定义为 $C^{⁎}$-某些étalegroupoids的代数，由 $（{\mathfrak{大号}}^{-}，{\mathfrak{大号}}^{+}）$。我们研究的结构$C^{⁎}$-代数，并证明它们是通用单位唯一的 $C^{⁎}$-代数服从某些等距的典范生成器与由λ-图双系统的结构编码的投影之间的某些算符关系$（{\mathfrak{大号}}^{-}，{\mathfrak{大号}}^{+}）$。如果λ图双系统来自有限有向图的λ图系统，则关联的子位移为两侧拓扑马尔可夫位移$（{\mathrm{\Lambda }}_{\mathrm{一种}}，{\sigma }_{\mathrm{一种}}）$通过图的过渡矩阵A以及相关联的$C^{⁎}$-代数 ${\mathcal{Ø}}_{{\mathfrak{大号}}^{-}}^{+}$ 与Cuntz-Krieger代数同构 ${\mathcal{Ø}}_{\mathrm{一种}}$，而另一个 $C^{⁎}$-代数 ${\mathcal{Ø}}_{{\mathfrak{大号}}^{+}}^{-}$ 与交叉乘积同构 $C^{⁎}$-代数 $C（{\mathrm{\Lambda }}_{\mathrm{一种}}）{⋊}_{{\sigma }_{\mathrm{一种}}^{⁎}}\mathbb{ž}$ 可交换的 $C^{⁎}$-代数 $C（{\mathrm{\Lambda }}_{\mathrm{一种}}）$ 函数在移位空间上的分布 ${\mathrm{\Lambda }}_{\mathrm{一种}}$ 自同构对两类拓扑马尔可夫位移的影响 ${\sigma }_{\mathrm{一种}}^{⁎}$ 由顺势同态引起 ${\sigma }_{\mathrm{一种}}$。这种现象表明Cuntz-Krieger代数之间是对偶的${\mathcal{Ø}}_{\mathrm{一种}}$ 和划线产品 $C^{⁎}$-代数 $C（{\mathrm{\Lambda }}_{\mathrm{一种}}）{⋊}_{{\sigma }_{\mathrm{一种}}^{⁎}}\mathbb{ž}$。