State splitting, strong shift equivalence and stable isomorphism of Cuntz-Krieger algebras

TL;DR

This paper demonstrates that strong shift equivalence of nonnegative matrices implies conjugacy of their associated stable Cuntz-Krieger algebras, using a functional analytic approach and graph-based bimodules.

Contribution

It establishes a new link between matrix equivalence and algebraic conjugacy of Cuntz-Krieger algebras, clarifying K-theoretic behavior and topological conjugacy.

Findings

Strong shift equivalence implies conjugate stable Cuntz-Krieger algebras.

Construction of imprimitivity bimodules from bipartite graphs elucidates algebraic behavior.

Topological conjugacy relates to algebraic equivalence of Cuntz-Krieger algebras with gauge actions.

Abstract

We prove that if two nonnegative matrices are strong shift equivalent, the associated stable Cuntz-Krieger algebras with generalized gauge actions are conjugate. The proof is done by a purely functional analytic method and based on constructing imprimitivity bimodule from bipartite directed graphs through strong shift equivalent matrices, so that we may clarify K-theoretic behavior of the stable conjugacy between the associated stable Cuntz-Krieger algebras. We also examine our machinery for the matrices obtained by state splitting graphs, so that topological conjugacy of the topological Markov shifts is described in terms of some equivalence relation of the Cuntz-Krieger algebras with canonical masas and the gauge actions without stabilization.