Classifiability of crossed products

Abstract

This survey studies C${}^{\ast }$-algebraic crossed products arising from topological dynamical systems with an eye toward their classifiability in the sense of the Elliott program. We introduce the crossed product construction for actions by discrete groups in full generality and then focus on commutative systems to establish some of the fundamental structural results: nuclearity of the crossed product is equivalent to amenability of the action, and in this setting, simplicity is equivalent to the combination of minimality and topological freeness. With these properties in place, the only condition left to translate into dynamical terms is tensorial absorption of the Jiang–Su algebra $\mathcal{Z}$. We present the main dynamical tools known to obtain $\mathcal{Z}$-stability, both in the settings of amenable and non-amenable groups, and highlight the two conjectures that are believed to capture the full picture. The aim of the survey is to provide both an accessible entry point and a comprehensive reference on the classification of crossed products, describing the state of the art and a number of central open problems in the field.