发表机构
Chalmers University of Technology; University of Gothenburg（查尔姆斯理工大学; 哥德堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本综述系统研究拓扑动力系统交叉积在 Elliott 纲领下的可分类性，总结核性、单性及 Z-稳定性的动力学刻画，并指出两个核心猜想与开放问题。
AI 中文摘要
本综述研究由拓扑动力系统产生的C*-代数交叉积，着眼于 Elliott 纲领意义上的可分类性。我们全面介绍离散群作用的交叉积构造，然后聚焦于交换系统以建立一些基本的结构性结果：交叉积的核性等价于作用的顺从性，并且在此背景下，单性等价于极小性与拓扑自由性的结合。在这些性质就位后，唯一需要转化为动力学条件的性质是 Jiang-Su 代数 $\mathcal{Z}$ 的张量吸收。我们介绍了已知的获得 $\mathcal{Z}$-稳定性的主要动力学工具，既适用于顺从群也适用于非顺从群，并强调了被认为能够刻画完整图景的两个猜想。本综述旨在为交叉积的分类提供一个易于入门的切入点以及全面的参考，描述该领域的最新进展和若干核心开放问题。
英文摘要
This survey studies C*-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 nonamenable 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.
Comments66 pages. Part of the EMS Series of Lectures in Mathematics "K-Theory and Operator Algebras", 2026
DOI：10.4171/elm/38
