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

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A λ-graph system L is a generalization of a finite labeled graph and presents a subshift. We will prove that the topological dynamical systems (X L₁,σ L₁) and (X L₂,σ L₂) for λ-graph systems L₁ and L₂ are continuously orbit equivalent if and only if there exists an isomorphism between the associated C*-algebras O L₁ and O L₂ keeping their commutative C*-subalgebras C(X L₁) and C(X L₂). It is also equivalent to the condition that there exists a homeomorphism from X L₁ to X L₂ intertwining their topological full inverse semigroups. In particular, one-sided subshifts XΛ₁ and XΛ₂ are λ-continuously orbit equivalent if and only if there exists an isomorphism between the associated C*-algebras OΛ₁ and OΛ₂ keeping their commutative C*-subalgebras C(XΛ₁) and C(XΛ₂).