Abstract Harmonic Analysis: New Frontiers

抽象谐波分析：新领域

• 批准号: RGPIN-2014-06356
• 负责人: Neufang, Matthias
• 金额: $ 2.04万
• 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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612178
• 关键词: Abstract; Harmonic; Analysis; New; Frontiers

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 our 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. In my recent work with Junge and Ruan, an important advance has been made: unifying and generalizing work of Ghahramani, Haagerup, Ruan, Spronk, Størmer, and myself, 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 by Hu, Ruan and myself. The latter is a fascinating object in its own right, e.g., in relation to the long-standing open problem of the duality between quantum group amenability and co-amenability. As shown with my former student Kalantar, 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, Kalantar and I have used our representation to build a functor from LC quantum groups to LC groups that preserves, e.g., compactness and discreteness. Numerous aspects of this assignment are yet to be studied; e.g., the possibility to express commutativity and co-commutativity via adjoints of this functor. Our construction leads to new invariants for LC quantum groups, in particular 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 yields an intriguing new class of quantum channels - an exciting connection with quantum information. Moreover, in ongoing work with Kalantar and Ruan, we determine the structure of the fixed point sets of these channels via a crossed product formula, thus obtaining a description of non-commutative Poisson boundaries. A further crucial feature of our representation is that it extends to large classes of bidual algebras, establishing an important link to topological centre problems, and to the famous Kadison-Singer Problem which has just been solved in 2013! We plan to tackle the topological centre problem for the Fourier-Stieltjes algebra, raised by Lau, via my factorization method: a very promising approach given, e.g., our recent solution of the Ghahramani-Lau conjecture on this question for the measure algebra. Related projects concern topological centres for group actions, and a tensor product version of topological centres.

Kustermans-Vaes在2000年提出的年轻的局部紧量子群理论，将经典的庞特里亚金对偶扩展到一个包含LC群和数学物理中出现的重要变形代数的范畴。我最近的工作有助于LC量子群谐波分析的快速发展，进而产生了许多有趣的新研究项目。因此，我们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Arens（IR）正则性。我们的出发点是以下基本问题。虽然抽象C*-代数的元素可以被看作是希尔伯特空间上的有界算子，但抽象调和分析的中心对象没有这样的表示，例如，群代数及其乘子代数，测度代数因此，在这种情况下构造一个表示模型是非常有意义的，更一般地说，对于LC量子群上的相应代数也是如此。在我最近与Junge和Ruan的工作中，取得了一个重要的进展：统一和推广了Ghahramani，Haagerup，Ruan，Spronk，Størmer和我自己的工作，我们已经为量子群代数的完全有界（右）乘子开发了一个表示模型，对于任何LC量子群。乘子代数可以通过与量子群相关联的迹类算子的卷积代数的自然作用来描述，正如胡，阮和我自己所介绍和研究的那样。后者本身就是一个迷人的对象，例如，关于量子群顺从性和协顺从性之间的对偶性这一长期存在的公开问题。正如我以前的学生Kalantar所示，迹类算子空间允许两个“对偶”乘积-卷积和逐点乘积的量子版本-通过一个可以被视为张量反对易关系的公式联系起来。这可能是发展超越量子群的对偶理论的起点。此外，Kalantar和我已经使用我们的表示来建立一个从LC量子群到LC群的函子，它保留了，例如，紧凑性和离散性。这项任务的许多方面还有待研究;例如，通过这个函子的伴随来表达交换性和余交换性的可能性。我们的建设导致新的LC量子群的不变量，特别是广义海森堡的bicharacters，其明确的计算是一项重要的任务，并可能，类似的精神K-理论，形成一个步骤的LC量子群的分类：一个程序的巨大的潜在影响。我们的表示也产生了一个有趣的新类量子通道-一个令人兴奋的量子信息的连接。此外，在与Kalantar和阮正在进行的工作中，我们通过交叉乘积公式确定了这些通道的不动点集的结构，从而获得了非交换泊松边界的描述。我们的表示的另一个重要特征是，它扩展到大类的双对偶代数，建立了一个重要的链接到拓扑中心问题，并在2013年刚刚解决了著名的Kadison-Singer问题！我们计划通过我的因式分解方法来解决Lau提出的Fourier-Stieltjes代数的拓扑中心问题：这是一种非常有前途的方法，例如，我们最近的解决方案的Ghahramani-Lau猜想这个问题的测度代数。相关的项目涉及群体作用的拓扑中心，以及拓扑中心的张量积版本。