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

Abstract

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

22 pages

References in corpus (1)

• Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras