A one-sided shift of finite type $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ 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 $\unicode[STIX]{x1D70F}_{A}$ on ${\mathcal{O}}_{A}$. On the other hand, $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines a groupoid ${\mathcal{G}}_{A}$ together with a certain homomorphism $\unicode[STIX]{x1D716}_{A}$ on ${\mathcal{G}}_{A}$. We show that each of these two sets of data completely characterizes the one-sided conjugacy class of $\mathsf{X}_{A}$. 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 provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.

中文翻译：

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CUNTZ-KRIEGER 代数和有限型移位及其 GROUPOID 的单边共轭

有限类型的单边移位$(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$一方面确定了 Cuntz-Krieger 代数${\mathcal{O}}_{A}$具有杰出的阿贝尔子代数${\mathcal{D}}_{A}$和某个完全正面的地图$\unicode[STIX]{x1D70F}_{A}$在${\mathcal{O}}_{A}$. 另一方面，$(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$确定一个groupoid${\mathcal{G}}_{A}$加上一定的同态$\unicode[STIX]{x1D716}_{A}$在${\mathcal{G}}_{A}$. 我们表明，这两组数据中的每一个都完全表征了单边共轭类$\mathsf{X}_{A}$. 这加强了 Cuntz 和 Krieger 的结果。我们还展示了一个有限类型的两个不可约移位的例子，它们最终是共轭的，但不是共轭的。这为松本关于最终共轭是否意味着共轭的问题提供了否定的答案。