Orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras

Abstract

We prove that one-sided topological Markov shifts (X A , σ A ) and (X B , σ B ) for matrices A and B with entries in {0, 1} are continuously orbit equivalent and there exists an isomorphism between the Cuntz-Krieger algebras O A and O B keeping their commutative C*-subalgebras C(X A ) and C(X B ). The part (and hence the only if part) above is 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 O A preserving the commutative C * -algebra C(X A ).