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Operator Algebras of Product Systems

产品系统的算子代数

基本信息

批准号：
EP/T02576X/1
负责人：
Evgenios Kakariadis
金额：
$ 3.21万
依托单位：
Newcastle University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2020
资助国家：
英国
起止时间：
2020 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT02576X%2F1
关键词：
Operator Algebras Product Systems

项目摘要

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.

在20世纪30年代，冯·诺依曼建议将算子代数作为量子力学系统方法的有效工具，将海森堡和薛定谔的观点联系起来。他的愿景已经发展成为一个坚实的框架，通过将表示论扩展到无限维来研究可逆变换，从而结合了代数和分析的工具。在这方面的努力的一个中心方面是确定不变量，可以有效地分类的C*-代数的一个可能的非交换状态空间上的一组行动。这样的结构已经出现在默里和冯·诺依曼的因子分类工作中。它们丰富的结构激发了对它们本身的研究，并激发了一般理论中的许多方向，因为它们为识别和测试关键结构提供了重要的实验设备。在过去的30年里，人们做出了巨大的努力来扩展我们对不可逆变换的理解。在一个变量的情况下，这导致了生产的运营商代数有关的结构非常重要的数学，如图，随机矩阵或分析品种。然而，对于更复杂的半群变换知之甚少，直到最近才通过许多专家的工作确定了正确的模型。这里的量子化不仅仅是群情形或单变量情形的扩展。这并不奇怪，因为半群的类太大了。因此，需要发明新的方法来进行这一层次的彻底分析。在当前的项目中，我们希望在可接受（和可接受控制）的转换背景下工作。它们形成了一个相当广泛的类别，涵盖了大量以前的构造，例如与阿贝尔格和高阶图有关的构造。我们的主要目标是研究其相关的算子代数的性质，从而促进他们的一个关键对象在C*-理论作为一个来源的例子，反例和applications.There有三个基本问题的基础上有效的分析有关Arveson的计划，埃利奥特的计划和Laca-Neshveyev-Raeburn计划，我们计划在这里追求。在16个月的时间里，将在PI对国际研究中心的四次访问中对这些问题进行调查。