Abstract Harmonic Analysis: New Frontiers

点击翻译按钮获取中文摘要

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.