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Positivity of Matui's HK Conjecture for AF Groupoids

Rafael P. Lima

TL;DR

The paper proves that for AF groupoids $G$, the HK conjecture map between $K_0(C^*(G))$ and the zeroth homology $H_0(G)$ is an order isomorphism with an explicit, trace-based formula $ u([p]_0)=[ ext{tr}_{oldsymbol{ au}_n}(p)]$, confirming positivity preservation and identifying $ u([1_V]_0)=[1_V]$. This explicit correspondence yields a practical tool for computing K-theory via homology and extends FKPS’s results to ordered group isomorphisms beyond compact unit spaces. The authors apply this HK isomorphism to Deaconu-Renault groupoids, deriving a Brown-type criterion for AF embeddability of $C^*(oldsymbol{ rak G})$ in terms of a homology condition involving $ au_*$, and they exhibit a commutative diagram linking $K_0$ and $H_0$ through skew-products and crossed products. The framework unifies and extends known AF-embeddability results for graph algebras, crossed products, topological graph algebras, and Cuntz–Pimsner algebras, providing concrete, computable criteria grounded in groupoid homology.

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.

Positivity of Matui's HK Conjecture for AF Groupoids

TL;DR

The paper proves that for AF groupoids , the HK conjecture map between and the zeroth homology is an order isomorphism with an explicit, trace-based formula , confirming positivity preservation and identifying . This explicit correspondence yields a practical tool for computing K-theory via homology and extends FKPS’s results to ordered group isomorphisms beyond compact unit spaces. The authors apply this HK isomorphism to Deaconu-Renault groupoids, deriving a Brown-type criterion for AF embeddability of in terms of a homology condition involving , and they exhibit a commutative diagram linking and through skew-products and crossed products. The framework unifies and extends known AF-embeddability results for graph algebras, crossed products, topological graph algebras, and Cuntz–Pimsner algebras, providing concrete, computable criteria grounded in groupoid homology.

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.
Paper Structure (20 sections, 46 theorems, 120 equations)

This paper contains 20 sections, 46 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Inductive limits
  4. AF groupoids
  5. Homology in algebra
  6. Homology for groupoids
  7. Homology for AF groupoids
  8. Main result
  9. Application to Deaconu-Renault groupoids
  10. Deaconu-Renault groupoids
  11. Skew-products and crossed products
  12. The Commutative Diagram
  13. Part 1 of the diagram
  14. Part 2 of the diagram
  15. AF embeddability for Deaconu-Renault groupoids
  16. ...and 5 more sections

Key Result

Theorem 1

Let $G$ be an AF groupoid, written as the inductive limit $G = \lim_\rightarrow G_n$ of elementary groupoids. Then, for every $n$, there exists a locally compact, Hausdorff, second countable, totally disconnected space $Y_n$, and there is a surjective local homeomorphism $\sigma_n: G_n^{(0)} \righta

Theorems & Definitions (100)

  • Conjecture 1.1: Matui's HK conjecture
  • Theorem 1
  • Definition
  • Theorem 2
  • Definition
  • Theorem 1.2
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 90 more