AF Embeddability of the C*-Algebra of a Deaconu-Renault Groupoid

TL;DR

This paper characterizes when the C*-algebras of Deaconu-Renault groupoids, arising from surjective local homeomorphisms on certain topological spaces, are AF embeddable, generalizing previous results for graphs and crossed products.

Contribution

It provides a new condition for AF embeddability of these C*-algebras and explicitly relates homology groups to K-theory, extending prior theorems with explicit formulas.

Findings

Established a criterion for AF embeddability based on the surjective local homeomorphism.

Derived an explicit isomorphism between homology groups and K-theory that preserves positivity.

Generalized existing theorems for graph and crossed product C*-algebras.

Abstract

We study Deaconu-Renault groupoids corresponding to surjective local homeomorphisms on locally compact, Hausdorff, second countable, totally disconnected spaces, and we characterise when the C*-algebras of these groupoids are AF embeddable. Our main result generalises theorems in the literature for graphs and for crossed products of commutative C*-algebras by the integers. We give a condition on the surjective local homeomorphism that characterises the AF embeddability of the C*-algebra of the associated Deaconu-Renault groupoid. In order to prove our main result, we analyse homology groups for AF groupoids, and we prove a theorem that gives an explicit formula for the isomorphism of these groups and the corresponding K-theory. This isomorphism generalises Farsi, Kumjian, Pask, Sims (M\"unster J. Math, 2019) and Matui (Proc. Lond. Math. Soc, 2012), since we give an explicit formula for the isomorphism and we show that it preserves positive elements.