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。与此同时，从一开始的主题，预计运营商代数形成非常有趣的结构本身的权利，并将有应用程序的酉表示群和算子理论在希尔伯特空间。事实上，更多的东西被证明是真的。经过一些戏剧性的和意想不到的发展，算子代数理论已经确立了自己作为一个非常活跃的和高度跨学科的研究领域。算子代数不仅如最初设想的那样与量子物理、表示论和算子理论有着密切的联系，而且如今在各种数学学科如泛函分析、代数、几何群论、几何、拓扑或动力系统中有着深远的应用。C*-代数是算子代数中最重要的一类，它被定义为范数闭的，Hilbert空间上有界线性算子的自伴代数。正如在数学的许多领域，C*-代数理论的进展与有趣的和有启发性的例子的发现齐头并进，最突出的是群C*-代数和附着到动力系统的C*-代数，所谓的交叉积。本研究项目的主要研究对象是半群C*-代数，它们是群C*-代数的自然推广。我们的目标是分析半群C*-代数的结构，并利用这种结构作为工具，从几何群论的角度来研究群及其子半群。与此密切相关的是，本项目也旨在更好地理解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。我们的研究课题主要包括半群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、算子代数和动力系统中刚性现象的发现等。它位于数学的几个研究领域的接口，汇集了来自不同领域的专业知识。这就顺应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，数学界作为一个整体受益于积极和鼓舞人心的思想交流。

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.