Topological Conjugacy of Topological Markov Shifts and Cuntz-Krieger Algebras

摘要

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