Abstract Harmonic Analysis: New Frontiers

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，我们提议的主要目的是探索这一庞大计划的新领域：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。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，Stormer和我自己的工作，我们发展了一个量子群代数的完全有界(右)乘子的表示模型，对于任何LC量子群。乘子代数可以通过与量子群有关的迹类算符的卷积代数的自然作用来描述，这是由胡、阮和我介绍和研究的。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从性之间的二元性这一长期悬而未决的问题有关。正如我以前的学生Kalantar所展示的那样，迹类运算符的空间允许两个“对偶”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，Kalantar和我利用我们的表示构造了一个从LC量子群到LC群的函子，它保持了紧性和离散性等性质。这种赋值的许多方面还有待研究，例如，通过这个函子的伴随来表示交换性和余交换性的可能性。我们的构造导致了LC量子群的新的不变量，特别是推广了海森伯格的双特征，它的显式计算是一项重要的任务，它可能在精神上类似于K-理论，形成LC量子群分类的一步：一个具有巨大潜在影响的程序。我们的表示还产生了一类有趣的新量子通道-与量子信息的令人兴奋的连接。此外，在与Kalantar和Ruan正在进行的工作中，我们通过一个交叉积公式确定了这些通道的不动点集的结构，从而得到了非对易Poisson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双对偶代数，建立了到拓扑中心问题和著名的Kadison-Singer问题的重要联系，该问题在2013年刚刚得到解决！我们计划通过我的因式分解方法来解决Liu提出的Fourier-Stieltjes代数的拓扑中心问题：给出了一个非常有前途的方法，例如，我们最近解决了Ghahramani-Lau猜想关于这个问题的度量代数的问题。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。

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, Stormer, 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.