An étale equivalence relation on a configuration space arising from a subshift and related $C^{\ast }$-algebras

Abstract

A $\lambda$-graph bisystem $\mathcal{L}$ consists of two labeled Bratteli diagrams $({\mathcal{L}}^{-},{\mathcal{L}}^{+})$, that presents a two-sided subshift ${\Lambda }_{\mathcal{L}}$. We will construct a compact totally disconnected metric space with a shift homeomorphism consisting of two-dimensional configurations from a $\lambda$-graph bisystem. The configuration space has a certain étale AF-equivalence relation written $R_{\mathcal{L}}$ with a natural shift homeomorphism ${\sigma }_{\mathcal{L}}$ coming from the shift homeomorphism ${\sigma }_{{\Lambda }_{\mathcal{L}}}$ on the subshift ${\Lambda }_{\mathcal{L}}$. The equivalence relation $R_{\mathcal{L}}$ yields an AF-algebra ${\mathcal{F}}_{\mathcal{L}}$ with an automorphism ${\rho }_{\mathcal{L}}$ on it. We will study invariance of the étale equivalence relation $R_{\mathcal{L}}$, the groupoid ${\mathcal{G}}_{\mathcal{L}}=R_{\mathcal{L}}{⋊}_{{\sigma }_{\mathcal{L}}}\mathbb{Z}$ and the groupoid $C^{\ast }$-algebras $C^{\ast }(R_{\mathcal{L}})$, $C^{\ast }({\mathcal{G}}_{\mathcal{L}})$ under topological conjugacy of the presenting two-sided subshifts.

37 pages

References in corpus (2)

• Topological conjugacy of topological Markov shifts and Ruelle algebras
• $C^{\ast }$-algebras associated with two-sided subshifts