Rafael P. Lima | Searcharxiv

arXiv2025

Positivity of Matui's HK Conjecture for AF Groupoids

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.

math.OA↗

arXiv2023

Equivalence of definitions of AF groupoid

We prove the equivalence of two definitions of AF groupoid in the literature: one by Renault and the other by Farsi, Kumjian, Pask and Sims. In both definitions, an AF groupoid is an increasing union of more basic groupoids, called elementary groupoids. Surprisingly, the two definitions of elementary groupoid are not equivalent; they coincide if and only if the local homeomorphism that characterises them is a covering map.

math.OA↗