Almost elementariness and fiberwise amenability for étale groupoids

Abstract

In this paper, we introduce two new types of approximation properties for étale groupoids, almost elementariness and (ubiquitous) fiberwise amenability, inspired by Matui's and Kerr's notions of almost finiteness. In fact, we show that, in their respective scopes of applicability, both notions of almost finiteness are equivalent to the conjunction of our two properties. Our new properties stem from viewing étale groupoids as coarse geometric objects in the spirit of geometric group theory. Fiberwise amenability is a coarse geometric property of étale groupoids that is closely related to the existence of invariant measures on unit spaces and corresponds to the amenability of the acting group in a transformation groupoid. Almost elementariness may be viewed as a better dynamical analogue of the regularity properties of C * -algebras than almost finiteness, since, unlike the latter, the former may also be applied to the purely infinite case. To support this analogy, we show almost elementary minimal groupoids give rise to tracially Z-stable reduced groupoid C * -algebras. In particular, the C * -algebras of minimal second countable amenable almost finite groupoids in Matui's sense are Z-stable. Contents 1. Introduction 1 2. Preliminaries 8 3. Amenability of extended coarse spaces 11 4. Coarse geometry of étale groupoids 15 5. Fiberwise amenability 21 6. Almost elementary étale groupoids and groupoid strict comparison 30 7. Almost elementariness and almost finiteness 39 8. Small boundary property and a nesting form of almost elementariness 46 9. Tracial Z-stability 56 References 67This is one reason why we decided not to use the word "finite" in the name of our new notion, after a helpful suggestion of George Elliott to the second author.

Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces

“…Interestingly, the notion of comparison also appears in the classification programme of C*-algebras [46,48,47,57]. For instance, transformation groupoids of free minimal actions of groups with subexponential growth and elementary amenable groups on the Cantor space have comparison [18,47].…”

On tracial $\mathcal Z$-stability of simple non-unital C*-algebras

“…On the other hand, G| K is a minimal, almost finite, σ-compact étale groupoid with compact totally disconnected unit space K. As K is clopen in G (0) , K has no isolated points as well. By [30,Corollary 9.11], we deduce that C * r (G| K ) is a unital simple and tracially Z-stable C * -algebra. Moreover, C * r (G| K ) has real rank zero and stable rank one (see [43,2]).…”

“…For the class of simple and unital C * -algebras, Hirshberg and Orovitz introduced the notion of tracial Z-stability which corresponds to a weakened form of tensorial absorption of the Jiang-Su algebra Z [20]. The weaker nature of this notion makes it easier to verify than Zstability in many examples from dynamical systems (see for instance [22,30,24,23,36,33]). Despite its apparent weaker form, it turns out to be equivalent to Z-stability under the presence of nuclearity.…”

Tracial approximate divisibility and stable rank one

“…Again, using projectivity of C 0 ((0, 1]) ⊗ M 2 , there is a homomorphism ψ : C 0 ((0, 1]) ⊗ M 2 → l ∞ (A) ⊂ l ∞ (N ) that lifts ψ. We may represent ψ by a sequence of homomorphisms From a recent result of Ma and Wu in [29] on groupoid C * -algebras, let us restate the following.…”