Locally Compact Quantum Groups. A von Neumann Algebra Approach

Abstract

In this paper, we give an alternative approach to the theory of locally compact quantum groups, as developed by Kustermans and Vaes. We develop the theory completely within the von Neumann algebra framework. At various points, we also do things differently. We have a different treatment of the antipode. We obtain the uniqueness of the Haar weights in an early stage. We take advantage of this fact when deriving the other main results in the theory. We also give a slightly different approach to duality. Finally, we collect, in a systematic way, several important formulas. In an appendix, we indicate very briefly how the $C^{\ast }$-approach and the von Neumann algebra approach eventually yield the same objects.

References in corpus (3)

• Notes on Compact Quantum Groups
• The Haar measure on some locally compact quantum groups
• The multiplicative unitary as a basis for duality

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