TOPOLOGICAL CONJUGACY OF TOPOLOGICAL MARKOV SHIFTS AND CUNTZ-KRIEGER ALGEBRAS

For an irreducible non-permutation matrix A, the triplet (O-A, D-A, rho(A)) for the Cuntz-Krieger algebra O-A, its canonical maximal abelian C*-subalgebra 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 (O-A, D-A, rho(A)) and (O-B, 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.