Geodesic
Topological Conjugacy of Topological Markov Shifts and Cuntz-Krieger Algebras
Documenta mathematica, Tome 22 (2017), pp. 873-915.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

For an irreducible non-permutation matrix $A$, the triplet $(\Cal{O}_A,\Cal{D}_A,\rho^A)$ for the Cuntz-Krieger algebra $\Cal{O}_A$, its canonical maximal abelian $C^\ast$-subalgebra $\Cal{D}_A$, 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 $(\Cal{O}_A,\Cal{D}_A,\rho^A)$ and $(\Cal{O}_B,\Cal{D}_B,\rho^B)$ 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.
Classification : 46L55, 46L35, 37B10
Keywords: topological Markov shifts, topological conjugacy, strong shift equivalence, Cuntz-Krieger algebras, K-theory, gauge action 
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     author = {Matsumoto, Kengo},
     title = {Topological {Conjugacy} of {Topological} {Markov} {Shifts} and {Cuntz-Krieger} {Algebras}},
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     url = {http://geodesic.mathdoc.fr/item/DOCMA_2017__22__a28/}
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Matsumoto, Kengo. Topological Conjugacy of Topological Markov Shifts and Cuntz-Krieger Algebras. Documenta mathematica, Tome 22 (2017), pp. 873-915. http://geodesic.mathdoc.fr/item/DOCMA_2017__22__a28/