Topological Conjugacy of Topological Markov Shifts and Cuntz--Krieger Algebras

For an irreducible non-permutation matrix $A$, the triplet $(\OA,\DA,\rho^A)$ for the Cuntz-Krieger algebra $\OA$, its canonical maximal abelian $C^*$-subalgebra $\DA$, 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 $(\OA,\DA,\rho^A)$ and $(\OB,\DB,\rho^B)$ 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.

2010 Mathematics Subject Classification: Primary 46L55; Secondary 46L35, 37B10.

Keywords and Phrases: Topological Markov shifts, topological conjugacy, strong shift equivalence, Cuntz--Krieger algebras, K-theory, gauge action.

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