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 $C^*$-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.