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Subshifts, $λ$-graph bisystems and $C^*$-algebras

We introduce a notion of $λ$-graph bisystem. It consists of a pair $({\frak L}^-, {\frak L}^+)$ of two labeled Bratteli diagrams ${\frak L}^-, {\frak L}^+$ over alphabets $Σ^-, Σ^+$, respectively, and satisfy certain compatibility condition of their labeling on edges. 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 a notion of properly strong shift equivalence on symbolic matrix bisystems 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 $({\frak L}^-, {\frak L}^+)$ yields a pair of $C^*$-algebra written ${\mathcal{O}_{{\frak L}^-}^+}, {\mathcal{O}_{{\frak L}^+}^-}$. We show that the $C^*$-algebras are universal unique $C^*$-algebras subject to certain operator relations among canonical generators encoded by $λ$-graph bisystem $({\frak L}^-, {\frak L}^+).$ 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 $(Λ_A, σ_A)$ by its transition matrix $A$ of the graph, and the associated $C^*$-algebra ${\mathcal{O}_{{\frak L}^-}^+}$ is isomorphic to ${\mathcal{O}}_A,$ whereas the other $C^*$-algebra ${\mathcal{O}_{{\frak L}^+}^-}$ is isomorphic to $C(Λ_A)\rtimes_{σ_A^*}\mathbb{Z}$ of the commutative $C^*$-algebra $C(Λ_A)$ on $Λ_A$ by the automorphism induced by the homeomorphism of the shift $σ_A.$ This phenomena shows a duality between ${\mathcal{O}}_A$ and $C(Λ_A)\rtimes_{σ_A^*}\mathbb{Z}$.