On the AF embeddability of crossed products of AF algebras by the integers

TL;DR

This paper characterizes when crossed products of AF algebras by integers can be embedded into AF algebras, linking AF embeddability to stable finiteness and K-theoretic properties.

Contribution

It provides a K-theoretic criterion for AF embeddability of crossed products and shows how embeddings induce rationally injective maps on K_0.

Findings

Crossed product is AF embeddable iff it is stably finite.

AF embedding can be chosen to induce a rationally injective K_0 map.

K-theoretic characterization simplifies understanding AF embeddability.

Abstract

It is shown that if A is an AF algebra then a crossed product of A by the integers can be embedded into an AF algebra if and only if the crossed product is stably finite. This equivalence follows from a simple K-theoretic characterization of AF embeddability. It is then shown that if a crossed product of an AF algebra by the integers is AF embeddable then the AF embedding can be chosen in such a way as to induce a rationally injective map on K_0 of the crossed product.