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

TL;DR

This paper establishes a deep connection between orbit equivalence of one-sided subshifts and isomorphisms of their associated C*-algebras, providing new characterizations in symbolic dynamics and operator algebras.

Contribution

It proves that orbit equivalence of subshifts corresponds precisely to isomorphisms of their C*-algebras that preserve certain subalgebras, linking dynamical and algebraic structures.

Findings

Orbit equivalence characterized by C*-algebra isomorphisms

Homeomorphisms intertwining topological full inverse semigroups

Equivalent conditions for λ-continuous orbit equivalence

Abstract

A $\lambda$-graph system ${\frak L}$ is a generalization of a finite labeled graph and presents a subshift. We will prove that the topological dynamical systems $(X_{{\frak L}_1},\sigma_{{\frak L}_1})$ and $(X_{{\frak L}_2},\sigma_{{\frak L}_2})$ for $\lambda$-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 $X_{\Lambda_1}$ and $X_{\Lambda_2}$ are $\lambda$-continuously orbit equivalent if and only if there exists an isomorphism between the associated $C^*$-algebras ${\Cal O}_{\Lambda_1}$ and ${\Cal O}_{\Lambda_2}$ keeping their commutative $C^*$-subalgebras $C(X_{\Lambda_1})$ and $C(X_{\Lambda_2})$.