Elementary amenability and almost finiteness

TL;DR

The paper proves that free continuous actions of countably infinite elementary amenable groups on finite-dimensional compact spaces are almost finite, leading to classification results for their crossed product C*-algebras.

Contribution

It establishes almost finiteness for a broad class of group actions and derives classification results for the associated crossed product C*-algebras.

Findings

Actions are almost finite

Crossed products are $\\mathcal{Z}$-stable

Crossed products are classified by Elliott invariant

Abstract

We show that every free continuous action of a countably infinite elementary amenable group on a finite-dimensional compact metrizable space is almost finite. As a consequence, the crossed products of minimal such actions are $\mathcal{Z}$-stable and classified by their Elliott invariant.