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

Kevin Aguyar Brix; Toke Meier Carlsen

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

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arxiv 1712.00179 v2 pith:JF7VT3MQ submitted 2017-12-01 math.OA

Kevin Aguyar Brix , Toke Meier Carlsen This is my paper · ORCID

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keywords mathcalconjugacyfiniteone-sidedtypecertaincompletelyconjugate

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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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Subshift semigroups
math.OA 2019-08 accept novelty 7.0 of 10

For every subshift, the Matsumoto and Carlsen-Matsumoto C*-algebras are realized as groupoid C*-algebras from the inverse hull of the language semigroup, and this universal groupoid is amenable.

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