Duality, cohomology, and geometry of locally compact quantum groups

作者：Kalantar Mehrdad^*; Neufang Matthias

来源：Journal of Mathematical Analysis and Applications, 2013, 406(1): 22-33.

DOI：10.1016/j.jmaa.2013.04.024

• Locally compact quantum groups
• Convolution algebras
• Cohomology
• Amenability

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.

• 出版日期2013-10-1