A short note on Cuntz splice from a viewpoint of continuous orbit equivalence of topological Markov shifts

TL;DR

This paper introduces a method to construct an extended matrix that preserves the isomorphism class of Cuntz--Krieger algebras while flipping the sign of their determinants, linking algebraic invariants to continuous orbit equivalence of topological Markov shifts.

Contribution

It provides a straightforward way to relate Cuntz--Krieger algebra isomorphisms to continuous orbit equivalence via a matrix construction that alters the determinant sign.

Findings

Constructed an extended irreducible matrix with isomorphic Cuntz--Krieger algebras.

Established the equivalence between algebra isomorphism and continuous orbit equivalence.

Linked determinant sign change to orbit equivalence classes.

Abstract

Let $A$ be an $N\times N$ irreducible matrix with entries in $\{0,1\}$. We present an easy way to find an $(N+3)\times (N+3)$ irreducible matrix $\bar{A}$ with entries in $\{0,1\}$ such that their Cuntz--Krieger algebras ${\mathcal{O}}_A$ and ${\mathcal{O}}_{\bar{A}}$ are isomorphic and $ \det(1 -A) = - \det(1-\bar{A}). $ As a consequence, we know that two Cuntz--Krieger algebras ${\mathcal{O}}_A$ and ${\mathcal{O}}_B$ are isomorphic if and only if the one-sided topological Markov shift $(X_A, \sigma_A)$ is continuously orbit equivalent to $(X_B, \sigma_B)$ or $(X_{\bar{B}}, \sigma_{\bar{B}}).$