Topological Conjugacy of Topological Markov Shifts and Cuntz–Krieger Algebras

For an irreducible non-permutation matrix A , the triplet (\mathcal{O}_A,\mathcal{D}_A,\rho^A) for the Cuntz–Krieger algebra \mathcal{O}_A , its canonical maximal abelian C^\ast -subalgebra \mathcal{D}_A , and its gauge action \rho^A is called the Cuntz–Krieger triplet. We introduce a notion of strong Morita equivalence in the Cuntz–Krieger triplets, and prove that two Cuntz–Krieger triplets (\mathcal{O}_A,\mathcal{D}_A,\rho^A) and (\mathcal{O}_B,\mathcal{D}_B,\rho^B) are strong Morita equivalent if and only if A and B are strong shift equivalent. We also show that the generalized gauge actions on the stabilized Cuntz–Krieger algebras are cocycle conjugate if the underlying matrices are strong shift equivalent. By clarifying K-theoretic behavior of the cocycle conjugacy, we investigate a relationship between cocycle conjugacy of the gauge actions on the stabilized Cuntz–Krieger algebras and topological conjugacy of the underlying topological Markov shifts.