Topological Conjugacy of Topological Markov Shifts and Cuntz–Krieger Algebras

Kengo Matsumoto

Documenta mathematica, Tome 22 (2017), pp. 873-915

Résumé

For an irreducible non-permutation matrix A, the triplet (OA​,DA​,ρA) for the Cuntz–Krieger algebra OA​, its canonical maximal abelian C∗-subalgebra DA​, and its gauge action ρA 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​,ρA) and (OB​,DB​,ρB) 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.

Détail

DOI : 10.4171/dm/581

Classification : 37B10, 46L35, 46L55
Mots-clés : K-theory, topological conjugacy, strong shift equivalence, Cuntz-Krieger algebras, topological Markov shifts, gauge action