  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 $C(X_{\Lambda_2})$.
