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

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 ${ϵ}_A$ 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.