Topological conjugacy of topological Markov shifts and Cuntz-Krieger algebras

TL;DR

This paper establishes a deep connection between the algebraic structure of Cuntz-Krieger algebras and the dynamical properties of topological Markov shifts, showing equivalences via Morita theory and cocycle conjugacy.

Contribution

It introduces a notion of strong Morita equivalence for Cuntz-Krieger triplets and proves its equivalence to strong shift equivalence of matrices, linking algebraic and dynamical classifications.

Findings

Strong Morita equivalence corresponds to strong shift equivalence of matrices.

Generalized gauge actions are cocycle conjugate under strong shift equivalence.

Cocycle conjugacy relates to topological conjugacy of Markov shifts.

Abstract

For an irreducible non-permutation matrix $A$, the triplet $({{\mathcal{O}}_A},{{\mathcal{D}}_A},\rho^A)$ for the Cuntz-Krieger algebra ${{\mathcal{O}}_A}$, its canonical maximal abelian $C^*$-subalgebra ${{\mathcal{D}}_A}$, 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 $({{\mathcal{O}}_A},{{\mathcal{D}}_A},\rho^A)$ and $({{\mathcal{O}}_B},{{\mathcal{D}}_B},\rho^B)$ 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.