This paper is a continuation of the paper, Matsumoto [‘Subshifts, $\lambda $ -graph bisystems and $C^*$ -algebras’, J. Math. Anal. Appl. 485 (2020), 123843]. A $\lambda $ -graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying a certain compatibility condition on their edge labeling. For any two-sided subshift $\Lambda $ , there exists a $\lambda $ -graph bisystem satisfying a special property called the follower–predecessor compatibility condition. We construct an AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ with shift automorphism $\rho _{\mathcal {L}}$ from a $\lambda $ -graph bisystem $({\mathcal {L}}^-,{\mathcal {L}}^+)$ , and define a $C^*$ -algebra ${\mathcal R}_{\mathcal {L}}$ by the crossed product . It is a two-sided subshift analogue of asymptotic Ruelle algebras constructed from Smale spaces. If $\lambda $ -graph bisystems come from two-sided subshifts, these $C^*$ -algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We present a simplicity condition of the $C^*$ -algebra ${\mathcal R}_{\mathcal {L}}$ and the K-theory formulas of the $C^*$ -algebras ${\mathcal {F}}_{\mathcal {L}}$ and ${\mathcal R}_{\mathcal {L}}$ . The K-group for the AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ is regarded as a two-sided extension of the dimension group of subshifts.

本文是论文 Matsumoto ['Subshifts, $\lambda $ -graph bisystems and $C^*$ -algebras', J. Math. 肛门。应用程序。485 (2020), 123843]。 $\lambda $ -graph 双系统由一对两个标记的 Bratteli 图组成，它们在边缘标记上满足一定的兼容性条件。 对于任何双边子移位 $\Lambda $ ，都存在一个 $\lambda $ -graph 双系统，满足称为从属-前驱兼容条件的特殊属性。我们从 $\lambda $ 构造一个具有移位自同构 $\rho _{\mathcal {L}}$ 的 AF 代数 ${\mathcal {F}}_{\mathcal {L}} $ -graph bisystem $({\mathcal {L}}^-,{\mathcal {L}}^+)$ ，并定义一个 $C^*$ -algebra ${\mathcal R}_{\mathcal {L} }$ 由交叉产品 。它是由 Smale 空间构造的渐近 Ruelle 代数的双边子移位模拟。如果 $\lambda $ -graph 双系统来自两侧子移位，则这些 $C^*$ - 代数在底层子移位的拓扑共轭下被证明是不变的。我们提出了 $C^*$ - 代数 ${\mathcal R}_{\mathcal {L}}$ 的简单条件和 $C^*$ -代数 ${\mathcal {F }}_{\mathcal {L}}$ 和 ${\mathcal R}_{\mathcal {L}}$ 。AF 代数 ${\mathcal {F}}_{\mathcal {L}}$ 的 K 群被视为子移位维群的两侧扩展。