Abstract Harmonic Analysis: New Frontiers

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，我们建议的主要目的是探索这个庞大项目的新领域：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。在我最近与Junge和阮的工作中，取得了一个重要的进展：统一和推广Ghahramani， Haagerup，阮，Spronk, st . rmer和我自己的工作，我们已经为任何LC量子群的量子群代数的完全有界（右）乘子开发了一个表示模型。乘子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，正如胡、阮和我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长期开放的问题有关。正如我以前的学生Kalantar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起来。这可能成为发展超越量子群的对偶理论的起点。此外，Kalantar和我已经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。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表示交换性和共交换性的可能性。我们的构造导致了LC量子群的新不变量，特别是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，形成了迈向LC量子群分类的一步：一个具有巨大潜在影响的程序。我们的表示还产生了一种有趣的新型量子通道——一种与量子信息的令人兴奋的连接。此外，在与Kalantar和阮正在进行的工作中，我们通过交叉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关键特征是它扩展到对偶代数的大类，建立了与拓扑中心问题的重要联系，以及2013年刚刚解决的著名的卡迪森-辛格问题！我们计划通过我的因式分解方法来解决由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，我们最近对度量代数的Ghahramani-Lau猜想的解决方案。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。

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.