Orbit equivalence of one-sided subshifts and the associated C^*-algebras

Abstract

A $λ$-graph system ${\frak L}$ is a generalization of a finite labeled graph and presents a subshift. We will prove that the topological dynamical systems $(X_{{\frak L}_1},σ_{{\frak L}_1})$ and $(X_{{\frak L}_2},σ_{{\frak L}_2})$ for $λ$-graph systems ${\frak L}_1$ and ${\frak L}_2$ are continuously orbit equivalent if and only if there exists an isomorphism between the associated $C^*$-algebras ${\Cal O}_{{\frak L}_1}$ and ${\Cal O}_{{\frak L}_2}$ keeping their commutative $C^*$-subalgebras $C(X_{{\frak L}_1})$ and $C(X_{{\frak L}_2})$. It is also equivalent to the condition that there exists a homeomorphism from $X_{{\frak L}_1}$ to $X_{{\frak L}_2}$ intertwining their topological full inverse semigroups. In particular, one-sided subshifts $X_{Λ_1}$ and $X_{Λ_2}$ are $λ$-continuously orbit equivalent if and only if there exists an isomorphism between the associated $C^*$-algebras ${\Cal O}_{Λ_1}$ and ${\Cal O}_{Λ_2}$ keeping their commutative $C^*$-subalgebras $C(X_{Λ_1})$ and $C(X_{Λ_2})$.