Advances in Abstract Harmonic Analysis

抽象谐波分析的进展

• 批准号: RGPIN-2020-06505
• 负责人: Neufang, Matthias
• 金额: $ 1.75万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=717779
• 关键词: Advances; Abstract; Harmonic; Analysis

The young theory of locally compact (LC) quantum groups, presented in 2000 by Kustermans-Vaes, extends classical Pontryagin duality to a category comprising both LC groups and important deformation algebras arising in mathematical physics. My recent work has contributed to the current rapid development of harmonic analysis on LC quantum groups - giving rise in turn to numerous intriguing novel research projects. The principal aim of my proposal is thus to explore new frontiers of this vast program: this will reveal both novel mathematical objects and approaches to open problems, through links with non-commutative ergodic theory and quantum information, as well as topological centres and Arens (ir)regularity. Our starting point is the following fundamental problem. While the elements of an abstract C*-algebra can be seen as bounded operators on a Hilbert space, there is no such representation for the central objects of abstract harmonic analysis, e.g., the group algebra and its multiplier algebra, the measure algebra. It is thus of great significance to construct a representation model in this setting - and, more generally, for the corresponding algebras over LC quantum groups. My research collaborators and I have made important advances: we have developed a representation model for the completely bounded (right) multipliers of the quantum group algebra, for any LC quantum group. The multiplier algebra can be described via a natural action of the convolution algebra of trace class operators associated with the quantum group, as introduced and studied as well by us. The latter is a fascinating object in its own right, e.g., in relation to the major open problem of the duality between quantum group amenability and co-amenability. As we have shown, the space of trace class operators admits two 'dual' products - quantum versions of convolution and pointwise product - linked by a formula that can be viewed as a tensorial anti-commutation relation. This may form the starting point to develop a duality theory beyond quantum groups. Moreover, we have used our representation to build a functor from LC quantum groups to LC groups that preserves, e.g., compactness and discreteness. Our construction leads to new invariants for LC quantum groups, generalizing Heisenberg's bicharacters, whose explicit calculation is an important task, and which may, similar in spirit to K-theory, form a step towards a classification of LC quantum groups: a program of great potential impact. Our representation also extends to large classes of bidual algebras, establishing important links to topological centre problems, and to the famous Kadison-Singer Problem, which has only been solved in 2015. We plan to tackle the topological centre problem for the Fourier-Stieltjes algebra via my factorization method. Related projects concern topological centres for group actions, and a tensor product version of topological centres.

由Kustermans-Vaes于2000年提出的局部紧（LC）量子群的年轻理论，将经典的庞特里亚金对偶扩展到包含LC群和数学物理中出现的重要变形代数的范畴。我最近的工作为当前LC量子群谐波分析的快速发展做出了贡献，从而引发了许多有趣的新研究项目。因此，我的建议的主要目的是探索这个庞大计划的新领域：这将揭示新的数学对象和开放问题的方法，通过与非交换遍历理论和量子信息的联系，以及拓扑中心和阿伦斯（ir）规则。我们的出发点是下面这个基本问题。虽然抽象C*-代数的元素可以看作是Hilbert空间上的有界算子，但抽象调和分析的中心对象，例如群代数及其乘子代数、测度代数，却没有这样的表示。因此，在这种情况下，更普遍地说，对于LC量子群上的相应代数，构建一个表示模型具有重要意义。我的研究合作者和我已经取得了重要的进展：我们已经为任何LC量子群的量子群代数的完全有界（右）乘数开发了一个表示模型。乘数代数可以通过与量子群相关的迹类算子的卷积代数的自然作用来描述，正如我们所介绍和研究的那样。后者本身就是一个迷人的对象，例如，与量子群可顺从性和共顺从性之间的对偶性这一主要开放问题有关。正如我们所展示的，迹类算子的空间允许两个“对偶”积——卷积的量子版本和点积——由一个可以被视为张量反对易关系的公式连接。这可能成为发展超越量子群的对偶理论的起点。此外，我们已经使用我们的表示建立了一个从LC量子群到LC群的函子，它保持了紧性和离散性。我们的构造导致了LC量子群的新不变量，推广了海森堡的双特征，其显式计算是一项重要任务，并且可能在精神上类似于k理论，形成了LC量子群分类的一步：一个具有巨大潜在影响的程序。我们的表示也扩展到对偶代数的大类，建立了与拓扑中心问题和著名的卡迪森-辛格问题的重要联系，该问题在2015年才得到解决。我们计划用我的分解方法来解决傅里叶-斯蒂尔杰代数的拓扑中心问题。相关项目涉及群体行动的拓扑中心，以及拓扑中心的张量积版本。