$C^{\ast }$-algebras associated with two-sided subshifts

Abstract

This paper is a continuation of the paper entitled "Subshifts, $\lambda$-graph bisystems and $C^{\ast }$-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 ${\mathcal{F}}_{\mathfrak{L}}$ with shift automorphism ${\rho }_{\mathfrak{L}}$ from a $\lambda$-graph bisystem $({\mathfrak{L}}^{-},{\mathfrak{L}}^{+})$, and define a $C^{\ast }$-algebra ${\mathcal{R}}_{\mathfrak{L}}$ by the crossed product ${\mathcal{F}}_{\mathfrak{L}}{⋊}_{{\rho }_{\mathfrak{L}}}\mathbb{Z}$. 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 $C^{\ast }$-algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We will present a simplicity condition of the $C^{\ast }$-algebra ${\mathcal{R}}_{\mathfrak{L}}$ and the K-theory formulas of the $C^{\ast }$-algebras ${\mathcal{F}}_{\mathfrak{L}}$ and ${\mathcal{R}}_{\mathfrak{L}}$. The K-group for the AF-algebra ${\mathcal{F}}_{\mathfrak{L}}$ is regarded as a two-sided extension of the dimension group of subshifts.

50 pages

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