Operator Algebras of Product Systems

在20世纪30年代，冯·诺伊曼建议将算子代数作为一种有效的工具，用于系统地研究量子力学，将海森堡和Schrödinger的观点联系起来。他的愿景已经发展成为一个坚实的框架，通过将表示理论扩展到无限维度，从而结合代数和分析的工具来研究可逆变换。这一努力的一个中心方面是识别不变量，这些不变量可以有效地对可能非交换状态空间上群作用的C*-代数进行分类。默里和冯·诺伊曼在对因素进行分类时已经出现了这样的结构。它们丰富的结构在其自身方面激发了研究，并启发了一般理论的许多方向，因为它们为识别和测试关键猜想提供了重要的实验装置。在过去的30年里，人们做出了巨大的努力，将我们的理解扩展到不可逆转的转变。在单变量的情况下，这导致了与数学中非常重要的结构（如图、随机矩阵或解析变量）相关的算子代数的产生。然而，对于更复杂的半群变换，人们所知甚少，直到最近才通过许多专家的工作确定了正确的模型。这里的量化不仅仅是群情况或单变量情况的扩展。这并不奇怪，因为半群的类别太大了。因此，需要发明新的方法，以便在这一层次上进行彻底的分析。在当前的项目中，我们希望在可修改（和可修改控制）转换的上下文中工作。它们形成了一个相当广泛的类，涵盖了大量先前的构造，例如与阿贝尔格和高秩图有关的构造。我们的主要目标是研究它们的相关算子代数的性质，从而将它们提升为C*理论中的一个关键对象，作为示例，反例和应用的来源。在对阿韦森的纲领、艾略特的纲领和拉卡-奈什维耶夫-雷伯恩的纲领进行有效分析的基础上，有三个基本问题，我们计划在这里进行探讨。这些问题将在PI在16个月期间四次访问国际研究中心期间进行调查。

In the 1930s, von Neumann suggested Operator Algebras as an efficient tool for the systematic approach to Quantum Mechanics that connects the viewpoints of Heisenberg and Schrödinger. His vision has evolved to a solid framework for studying reversible transformations by extending representation theory to infinite dimensions, and thus incorporating tools from both Algebra and Analysis. A central aspect in this endeavour is to identify invariants that can effectively classify the C*-algebra of a group action on a possibly noncommutative state space. Such constructs already appeared in the work of Murray and von Neumann for the classification of factors. Their rich structure has instigated research in its own respect and has inspired many directions in the general theory as they offer important experimental devices for identifying and testing key conjectures.In the past 30 years there has been a great effort to extend our understanding to irreversible transformations. In the one-variable case this has led to the production of operator algebras related to structures of great importance in Mathematics such as graphs, stochastic matrices or analytic varieties. However much less has been known for more involved semigroup transformations, and only recently the correct model has been identified through the work of many experts. The quantization here is more involved than just an extension of the group-case or the one-variable case. This is not a surprise as the class of semigroups is too vast. Thus new methods need to be invented for a thorough analysis at that level.In the current project we wish to work in the context of amenable (and amenably controlled) transformations. They form a rather broad class that covers a wide number of previous constructs, for example in relation to abelian lattices and higher-rank graphs. Our main goal is to study the properties of their related operator algebras and thus promote them to a key object in the C*-theory as a source for examples, counterexamples and applications.There are three fundamental questions at the basis of an effective analysis in relation to Arveson's Programme, Elliott's Programme and Laca-Neshveyev-Raeburn programme, which we plan to pursue here. Those will be investigated during four visits of the PI to international research centres over a period of 16 months.