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

Kengo Matsumoto

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arXiv · 1511.01193

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

Kengo Matsumoto

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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 $(X_A, σ_A)$ is continuously orbit equivalent to $(X_B, σ_B)$ or $(X_{\bar{B}}, σ_{\bar{B}}).$

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Kengo Matsumoto. 2016-05-07. A short note on Cuntz splice from a viewpoint of continuous orbit equivalence of topological Markov shifts. https://arxiv.org/abs/1511.01193

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