Positivity of Matui's HK Conjecture for AF Groupoids

TL;DR

This paper extends Matui's HK conjecture to AF groupoids, providing an explicit isomorphism between homology and K-theory, which aids in understanding the structure of associated C*-algebras.

Contribution

It offers a new explicit formula for the isomorphism and proves it is an order isomorphism, enhancing tools for analyzing AF groupoid C*-algebras.

Findings

The isomorphism between homology and K-theory is explicitly constructed.

The map is proven to be an order isomorphism.

Application to AF embeddability of Deaconu-Renault groupoid C*-algebras.

Abstract

In this paper we generalise an application of Matui's HK conjecture by Farsi, Kumjian, Pask, and Sims, that gives an isomorphism from the homology groups of AF groupoids to the corresponding K-theory. We give an explicit formula for this isomorphism, and we show that the map is an order isomorphism. Since homology groups are equipped with several useful techniques, this map can help us to understand the K-theory of the C*-algebra in more detail. To illustrate this, we apply the isomorphism to characterise the AF embeddability of the C*-algebra of Deaconu-Renault groupoids.