2002/05/27 by Alfons Van Daele, Van Daele, Alfons
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #math.OA
paper · pdf · doi:10.48550/arxiv.math/0205285
arxiv created 2002/05/27 · openalex publication_date 2002/05/27 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Any multiplier Hopf *-algebra with positive integrals gives rise to a locally compact quantum group (in the sense of Kustermans and Vaes). As a special case of such a situation, we have the compact quantum groups (in the sense of Woronowicz) and the discrete quantum groups (as introduced by Effros and Ruan). In fact, the class of locally compact quantum groups arising from such multiplier Hopf *-algebras is self-dual. The most important features of these objects are (1) that they are of a purely algebraic nature and (2) that they have already a great complexity, very similar to the general locally compact quantum groups. This means that they can serve as a good model for the general objects, at least from the purely algebraic point of view. They can therefore be used to study various aspects of the general case, without going into the more difficult technical aspects, due to the complicated analytic structure of a general locally compact quantum group. In this paper, we will first recall the notion of a multiplier Hopf *-algebra with positive integrals. Then we will illustrate how these \it algebraic quantum groups can be used to gain a deeper understanding of the general theory. An important tool will be the Fourier transform. We will also concentrate on certain actions and how they behave with respect to this Fourier transform. On the one hand, we will study this in a purely algebraic context while on the other hand, we will also pass to the Hilbert space framework.