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 L1-algebra of locally compact quantum groups. In particular, we obtain a criterion that distinguishes discreteness from the Radon-Nikodym property in this setting.

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Keywords Amenability, Cohomology, Convolution algebras, Locally compact quantum groups
Persistent URL dx.doi.org/10.1016/j.jmaa.2013.04.024
Journal Journal of Mathematical Analysis and Applications
Citation
Kalantar, M. (Mehrdad), & Neufang, T. (2013). Duality, cohomology, and geometry of locally compact quantum groups. Journal of Mathematical Analysis and Applications, 406(1), 22–33. doi:10.1016/j.jmaa.2013.04.024