Topological Conjugacy of Topological Markov Shifts and Cuntz–Krieger Algebras

Abstract

For an irreducible non-permutation matrix $A$, the triplet $({\mathcal{O}}_A,{\mathcal{D}}_A,{\rho }^A)$ 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.