Duality, Cohomology, and Geometry of Locally Compact Quantum Groups

TL;DR

This paper explores the cohomological and geometrical aspects of convolution algebras related to locally compact quantum groups, establishing new links between algebraic structures and topological properties, including a quantum amenability criterion.

Contribution

It introduces novel cohomological characterizations of quantum group properties and extends classical theorems to the quantum setting, especially regarding module structures and the Radon--Nikodym property.

Findings

Relation between convolution and pointwise multiplication in quantum groups

Cohomological criteria for topological properties of quantum groups

Quantum version of Hulanicki's theorem on amenability

Abstract

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.