AF $C^*$-algebras from non AF groupoids

Abstract

We construct ample groupoids from certain categories of paths, and prove that their $C^*$-algebras coincide with the continued fraction AF algebras of Effros and Shen. The proof relies on recent classification results for simple nuclear $C^*$-algebras. The groupoids are not principal. This provides examples of Cartan subalgebras in the continued fraction AF algebras that are isomorphic, but not conjugate, to the standard diagonal subalgebras.