AF $C^{\ast }$-algebras from non AF groupoids

Abstract

We construct ample groupoids from certain categories of paths, and prove that their $C^{\ast }$-algebras coincide with the continued fraction AF algebras of Effros and Shen. The proof relies on recent classification results for simple nuclear $C^{\ast }$-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.

version 4: final version, to appear in Transactions AMS; minor errors corrected

References in corpus (2)

• KMS states on the C*-algebras of non-principal groupoids
• On Bost-Connes type systems for number fields