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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759245
• 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.

2000年由Kustermann-Vaes提出的年轻的局部紧(LC)量子群理论，将经典的Pontryagin对偶推广到一个既包含LC群又包含数学物理中的重要形变代数的范畴。我最近的工作对目前LC量子群的谐波分析的快速发展做出了贡献-反过来又产生了许多有趣的新研究项目。因此，我提议的主要目的是探索这个庞大计划的新领域：这将通过与非对易遍历理论和量子信息以及拓扑中心和Arens(Ir)正则性的联系，揭示新的数学对象和解决开放问题的方法。我们的出发点是以下几个根本问题。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，在这种情况下构造一个表示模型具有重要意义--更广泛地说，对于LC量子群上相应的代数而言。我和我的研究合作者取得了重要进展：我们为任何LC量子群开发了量子群代数的完全有界(右)乘子的表示模型。乘子代数可以通过与量子群有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。后者本身就是一个令人着迷的对象，例如，与量子群顺从性和共顺性之间的二元性这一重大悬而未决的问题有关。正如我们已经证明的，迹类算子空间允许两个‘对偶’乘积--卷积和逐点积的量子版本--通过一个可以被视为张量反对易关系的公式联系在一起。这可能是发展超越量子群的对偶性理论的起点。此外，我们利用我们的表象构造了一个从LC量子群到LC群的函子，它保持了紧性和离散性等性质。我们的构造为LC量子群引入了新的不变量，推广了Heisenberg的双特征，它的显式计算是一项重要的任务，在本质上类似于K-理论，可能形成LC量子群分类的一步：一个具有巨大潜在影响的程序。我们的表示还扩展到大类双对偶代数，建立了与拓扑中心问题和著名的Kadison-Singer问题的重要联系，该问题直到2015年才被解决。我们计划用我的因式分解方法来解决Fourier-Stieltjes代数的拓扑中心问题。相关项目涉及群行动的拓扑中心，以及拓扑中心的张量积版本。