Duality, Cohomology, and Geometry of Locally Compact Quantum Groups

In this paper we study various convolution-type algebras associated with a locally compact quantum group from cohomological and geometrical points of view. The quantum group duality endows the space of trace class operators over a locally compact quantum group with two products which are operator versions of convolution and pointwise multiplication, respectively; we investigate the relation between these two products, and derive a formula linking them. Furthermore, we define some canonical module structures on these convolution algebras, and prove that certain topological properties of a quantum group, can be completely characterized in terms of cohomological properties of these modules. We also prove a quantum group version of a theorem of Hulanicki characterizing group amenability. Finally, we study the Radon--Nikodym property of the $L^1$-algebra of locally compact quantum groups. In particular, we obtain a criterion that distinguishes discreteness from the Radon--Nikodym property in this setting.