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

TL;DR

This paper constructs specific ample groupoids from path categories and shows their $C^*$-algebras match certain continued fraction AF algebras, revealing new non-conjugate Cartan subalgebras.

Contribution

It introduces a method to build groupoids whose $C^*$-algebras are the continued fraction AF algebras, providing new examples of Cartan subalgebras with unique properties.

Findings

Groupoids constructed from path categories yield the continued fraction AF algebras.

The $C^*$-algebras coincide with Effros and Shen's AF algebras.

Examples of non-conjugate Cartan subalgebras are provided.

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.