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arXiv · 1712.00179

Cuntz-Krieger algebras and one-sided conjugacy of shifts of finite type and their groupoids

Kevin Aguyar Brix·Toke Meier Carlsen
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Abstract

A one-sided shift of finite type $(X_A,\sigma_A)$ determines on the one hand a Cuntz-Krieger algebra $\mathcal{O}_A$ with a distinguished abelian subalgebra $\mathcal{D}_A$ and a certain completely positive map $\tau_A$ on $\mathcal{O}_A$. On the other hand, $(X_A,\sigma_A)$ determines a groupoid $\mathcal{G}_A$ together with a certain homomorphism $\epsilon_A$ on $\mathcal{G}_A$. We show that this data completely characterizes the one-sided conjugacy class of $X_A$. This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This answers a question of Matsumoto of whether eventual conjugacy implies conjugacy in the negative.

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Kevin Aguyar Brix, Toke Meier Carlsen. 2017-12-01. Cuntz-Krieger algebras and one-sided conjugacy of shifts of finite type and their groupoids. https://doi.org/10.1017/s1446788719000168

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