$\mathrm {C^*}$-algebras, groupoids and covers of shift spaces

Abstract:

To every one-sided shift space $\mathsf {X}$ we associate a cover $\widetilde {\mathsf {X}}$, 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 effective, we show that it is possible to recover the continuous orbit equivalence class of $\mathsf {X}$ from the pair $(\mathcal {O}_\mathsf {X}, C(\mathsf {X}))$, and the flow equivalence class of $\Lambda _\mathsf {X}$ from the pair $(\mathcal {O}_\mathsf {X}\otimes \mathbb {K}, C(\mathsf {X})\otimes c_0)$. In particular, continuous orbit equivalence implies flow equivalence for this class of shift spaces.