Topological conjugacy of topological Markov shifts and Cuntz-Krieger algebras

Abstract

For an irreducible non-permutation matrix $A$, the triplet $({\mathcal{O}_A},{\mathcal{D}_A},ρ^A)$ for the Cuntz-Krieger algebra ${\mathcal{O}_A}$, its canonical maximal abelian $C^*$-subalgebra ${\mathcal{D}_A}$, and its gauge action $ρ^A$ 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 $({\mathcal{O}_A},{\mathcal{D}_A},ρ^A)$ and $({\mathcal{O}_B},{\mathcal{D}_B},ρ^B)$ 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.