𝐶*-algebras, groupoids and covers of shift spaces

Abstract

To every one-sided shift space $\mathsf{X}$ we associate a cover $\tilde{\mathsf{X}}$, a groupoid $\mathcal{G}_{\mathsf{X}}$ and a $\mathrm{C^*}$-algebra $\mathcal{O}_{\mathsf{X}}$. We characterize one-sided conjugacy, eventual conjugacy and (stabilizer preserving) continuous orbit equivalence between $\mathsf{X}$ and $\mathsf{Y}$ in terms of isomorphism of $\mathcal{G}_{\mathsf{X}}$ and $\mathcal{G}_{\mathsf{Y}}$, and diagonal preserving $^*$-isomorphism of $\mathcal{O}_{\mathsf{X}}$ and $\mathcal{O}_{\mathsf{Y}}$. We also characterize two-sided conjugacy and flow equivalence of the associated two-sided shift spaces $\Lambda_{\mathsf{X}}$ and $\Lambda_{\mathsf{Y}}$ in terms of isomorphism of the stabilized groupoids $\mathcal{G}_{\mathsf{X}}\times \mathcal{R}$ and $\mathcal{G}_{\mathsf{Y}}\times \mathcal{R}$, and diagonal preserving $^*$-isomorphism of the stabilized $\mathrm{C^*}$-algebras $\mathcal{O}_{\mathsf{X}}\otimes \mathbb{K}$ and $\mathcal{O}_{\mathsf{Y}}\otimes \mathbb{K}$. Our strategy is to lift relations on the shift spaces to similar relations on the covers. Restricting to the class of sofic shifts whose groupoids are essentially principal, we find that the pair $(\mathcal{O}_{\mathsf{X}}, C(\mathsf{X}))$ remembers the continuous orbit equivalence class of $\mathsf{X}$ while the pair $(\mathcal{O}_{\mathsf{X}}\otimes \mathbb{K}, C(\mathsf{X})\otimes c_0)$ remembers the flow equivalence class of $\Lambda_{\mathsf{X}}$. In particular, continuous orbit equivalence implies flow equivalence for this class of shift spaces.