Advances in Abstract Harmonic Analysis

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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.