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Locally compact quantum groups. (English) Zbl 1034.46508
Summary: We propose a simple definition of a locally compact quantum group in reduced form. By the word ‘reduced’ we mean that we suppose the Haar weight to be faithful. So in fact we define and study an arbitrary locally compact quantum group, represented on the $$L^2$$-space of its Haar weight. For this locally compact quantum group we construct the antipode with polar decomposition. We construct the associated multiplicative unitary and prove that it is manageable in the sense of Woronowicz. We define the modular element and prove the uniqueness of the Haar weights. Following M. Enock and J. M. Schwartz [“Kac algebras and duality of locally compact groups” (Springer, Berlin) (1992; Zbl 0805.22003)], we construct the reduced dual, which will again be a reduced locally compact quantum group. Finally we prove that the second dual is canonically isomorphic to the original reduced locally compact quantum group, extending the Pontryagin duality theorem.
A short overview of the main results of this paper appeared in J. Kustermans and S. Vaes [C. R. Acad. Sci. Paris, Sér. I, Math. 328, 871–876 (1999; Zbl 0957.46037)].

##### MSC:
 46L65 Quantizations, deformations for selfadjoint operator algebras 46L89 Other “noncommutative” mathematics based on $$C^*$$-algebra theory 43A99 Abstract harmonic analysis 22D99 Locally compact groups and their algebras
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