$C^*$-algebras associated with two-sided subshifts

This paper is a continuation of the paper entitled "Subshifts, $\lambda$-graph bisystems and $C^*$-algebras", arXiv:1904.06464. A $\lambda$-graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying certain compatibility condition on their edge labeling. For any two-sided subshift $\Lambda$, there exists a $\lambda$-graph bisystem satisfying a special property called FPCC. We will construct an AF-algebra ${\mathcal{F}}_{\frak L}$ with shift automorphism $\rho_{\frak L}$ from a $\lambda$-graph bisystem $({\frak L}^-,{\frak L}^+)$, and define a $C^*$-algebra ${\mathcal R}_{\frak L}$ by the crossed product ${\mathcal{F}}_{\frak L}\rtimes_{\rho_{\frak L}}\mathbb{Z}$. 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 will present a simplicity condition of the $C^*$-algebra ${\mathcal R}_{\frak L}$ and the K-theory formulas of the $C^*$-algebras ${\mathcal{F}}_{\frak L}$ and ${\mathcal R}_{\frak L}$. The K-group for the AF-algebra ${\mathcal{F}}_{\frak L}$ is regarded as a two-sided extension of the dimension group of subshifts.