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Locally compact quantum groups in the von Neumann algebraic setting. (English) Zbl 1034.46067
The paper describes some aspects of the theory of locally compact quantum groups, extending the results obtained by the authors in [J. Kustermans and S. Vaes, Ann. Sci. Éc. Norm. Supér., IV. Sér. 33, 837–934 (2000; Zbl 1034.46508)]. In particular, the authors provide the definition of a locally compact quantum group in the von Neumann algebra setting, and then show how to obtain the corresponding \(C^*\)-algebra of a locally compact quantum group. A stronger version of the left invariance of the Haar weight is proven, and the duality of locally compact quantum groups is described.

MSC:
46L65 Quantizations, deformations for selfadjoint operator algebras
46L10 General theory of von Neumann algebras
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
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