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

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 $(X_{\bar{B}},{\sigma }_{\bar{B}}).$

8 pages, The title and the second half of the paper were changed from the preceding version

References in corpus (1)

• Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras