C*-algebras of semigroups and dynamical systems

算子代数的理论可以追溯到Murray， von Neumann， Gelfand和Naimark。最初的动机是为量子力学提供数学基础。同时，从一开始，我们就预料到算子代数可以形成非常有趣的结构，并且可以应用于Hilbert空间中群的酉表示和算子理论。事实上，更多的事实证明是正确的。经过一些戏剧性和意想不到的发展，算子代数理论已经成为一个非常活跃和高度跨学科的研究领域。不仅存在着——正如最初设想的那样——与量子物理、表示理论和算子理论的紧密联系，如今算子代数在各种数学学科中有着深远的应用，如泛函分析、代数、几何群论、几何、拓扑或动力系统。C*-代数是一类最重要的算子代数，它被定义为Hilbert空间上有界线性算子的范数闭自伴随代数。正如在数学的许多领域一样，C*-代数理论的发展伴随着有趣和有启发性的例子的发现，其中最突出的是群C*-代数和附在动力系统上的C*-代数，即所谓的交叉积。本课题的主要研究对象是半群C*-代数，它们是C*-代数群的自然推广。我们的目标是从几何群论的角度分析半群C*-代数的结构，并利用这种结构作为研究群及其子半群的工具。与此密切相关的是，该项目还旨在更好地理解C*代数与动力系统之间的相互作用。我们的项目处于当前研究的前沿。本文介绍了半群C*-代数、C*-代数的分类、C*-代数与符号动力学之间的相互作用以及算子代数和动力系统中刚性现象的发现等方面的最新进展。我们的研究项目的一个关键特点是其高度的跨学科性质。它位于几个数学研究领域的接口，汇集了来自不同领域的专业知识。这反映了数学中不同分支之间的相互作用变得越来越重要的趋势。因此，通过积极和鼓舞人心的思想交流，整个数学界受益。

The theory of operator algebras goes back to Murray, von Neumann, Gelfand and Naimark. The original motivation was to provide a mathematical foundation for quantum mechanics. At the same time, from the very beginning of the subject, it was anticipated that operator algebras form very interesting structures on their own right and will have applications to unitary representations of groups and operator theory in Hilbert space. Actually, much more turned out to be true. After some dramatic and unexpected developments, the theory of operator algebras has established itself as a very active and highly interdisciplinary research area. Not only do there exist - as initially envisioned - strong connections to quantum physics as well as representation theory and operator theory, operator algebras nowadays have far reaching applications in various mathematical disciplines like functional analysis, algebra, geometric group theory, geometry, topology or dynamical systems.One of the most important classes of operator algebras is given by C*-algebras, which are defined as norm-closed, self-adjoint algebras of bounded linear operators on a Hilbert space. As in many areas in mathematics, advances in the theory of C*-algebras went hand in hand with the discovery of interesting and illuminating examples, the most prominent ones being group C*-algebras and C*-algebras attached to dynamical systems, so-called crossed products.The main objects of study in this research project are given by semigroup C*-algebras, which are natural generalizations of group C*-algebras. Our goal is to analyse the structure of semigroup C*-algebras and to use this construction as a tool to study groups and their subsemigroups from the point of view of geometric group theory. Closely related to this, this project also aims at a better understanding of the interplay between C*-algebras and dynamical systems.Our project lies at the frontier of current research. We take up recent advances in semigroup C*-algebras, classification of C*-algebras, the interplay between C*-algebras and symbolic dynamics, as well as the discovery of rigidity phenomena in operator algebras and dynamical systems.One of the key characteristics of our research project is its high interdisciplinary character. It lies at the interface of several research areas in mathematics and brings together expertise from different fields. This takes up the trend in mathematics that interactions between different branches are becoming more and more important. Therefore, the mathematical community as a whole benefits through an active and inspiring exchange of ideas.