Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras

We will prove that one-sided topological Markov shifts $(X_A,{\sigma }_A)$ and $(X_B,{\sigma }_B)$ for matrices $A$ and $B$ with entries in $\{0,1\}$ are topologically orbit equivalent if and only if there exists an isomorphism between the Cuntz-Krieger algebras ${\text{Cal}O}_A$ and ${\text{Cal}O}_B$ keeping their commutative $C^{\ast }$-subalgerbas $C(X_A)$ and $C(X_B)$. It is also equivalent to the condition that there exists a homeomorphism from $X_A$ to $X_B$ intertwining their topological full groups. We will also study structure of the automorphisms of ${\text{Cal}O}_A$ keeping the commutative $C^{\ast }$-algebra $C(X_A)$.