$AF$-algebras and the tail-equivalence relation on Bratteli diagrams

We show that the $C^*$-algebra associated to the tail-equivalence relation on a Bratteli diagram, according to a procedure recently introduced by the first-named author and A. Lopes, is isomorphic to the $AF$-algebra of the diagram. More generally we consider an approximately proper equivalence relation $\mathcal{R}=\bigcup_{n\in\mathbb{N} }\mathcal{R}_n$ on a compact space $X$ for which the quotient maps $\pi_n\colon X\to X/\mathcal R_n$ are local homeomorphisms. We show that the algebra associated to $\mathcal{R}$ under the above-mentioned procedure is isomorphic to the groupoid $C^*$-algebra $C^*(\mathcal{R})$.