Topological full groups of ample groupoids with applications to graph algebras

Petter Nyland,Eduard Ortega

doi:10.1142/s0129167x19500186

Abstract

We study the topological full group of ample groupoids over locally compact spaces. We extend Matui’s definition of the topological full group from the compact to the locally compact case. We provide two general classes of étale groupoids for which the topological full group, as an abstract group, is a complete isomorphism invariant, hereby extending Matui’s Isomorphism Theorem. As an application, we study graph groupoids and their topological full groups, and obtain sharper results for this class. The machinery developed in this process is used to prove an embedding theorem for ample groupoids, akin to Kirchberg’s Embedding Theorem for [Formula: see text]-algebras. Consequences for graph [Formula: see text]-algebras and Leavitt path algebras are also spelled out. In particular, we improve on a recent embedding theorem of Brownlowe and Sørensen for Leavitt path algebras.

Highlights

The study of full groups in the setting of topological dynamics was initiated by Giordano, Putnam and Skau [27]
The topological full group is the subgroup of the full group consisting of those homeomorphisms which preserve the orbits in a continuous manner
We have chosen to break this direct step into two more parts in order to study when the groupoid can be recovered from the action of the topological full group on the unit space, as the groupoid of germs of this action. We find that such a groupoid of germs always embed into the groupoid we started with, and that they are isomorphic if and only if the subgroup in question is generated by enough bisections to cover the cIf Γ ≤ Homeo(X) and Λ ≤ Homeo(Y ) are groups of homeomorphisms, a spatial isomorphism between them is a homeomorphism φ : X → Y such that γ → φ ◦ γ ◦ φ−1 for γ ∈ Γ is a group isomorphism

Summary

Introduction

Background The study of (topological) full groups in the setting of topological dynamics was initiated by Giordano, Putnam and Skau [27]. Matui’s Isomorphism Theorem [49, Theorem 3.10] gives the equivalence of (2) and (4) above for the general class of ample effective Hausdorff minimal second countable groupoids over (compact) Cantor spaces (see Sec. 2.3 for definitions). This covers in particular graph groupoids of strongly connected finite graphs. We record a result on diagonal embeddings of AF-algebras in Corollary 11.27 Another consequence of Theorem E is that the topological full group GE , for any graph E as above, embeds into Thompson’s group V — since V is isomorphic to GE2. We remark that transformation groupoids arising from locally compact (noncompact) Cantor minimal systems are AF-groupoids, and GE2 -embeddable as well

Preliminaries

Topological notions

Stone duality

Etale groupoids

The Topological Full Group

The Groupoid of Germs

The Category of Spatial Groups

Spatial Realization Theorems

The class KF

The class KLCC

Reconstructing the space X

Isomorphism Theorems for Ample Groupoids