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Multiplier Hopf algebras imbedded in locally compact quantum groups. (English) Zbl 1226.16023
The paper under review is concerned with the relations between locally compact quantum groups in the sense of J. Kustermans and S. Vaes [Ann. Sci. Éc. Norm. Supér. (4) 33, No. 6, 837-934 (2000; Zbl 1034.46508)] and algebraic (or \(^*\)-algebraic) quantum groups introduced by the second named author [Adv. Math. 140, No. 2, 323-366 (1998; Zbl 0933.16043)]. Kustermans showed earlier that if \(\mathcal A\) is a \(^*\)-algebraic quantum group then it gives rise to a locally compact quantum group \(A\) (\(A\) is a certain completion of \(\mathcal A\)). Here a kind of a converse problem is studied: given a locally compact quantum group (either von Neumann algebraic or \(C^*\)-algebraic), which contains a regular multiplier Hopf algebra \(\mathcal A\), what properties does \(\mathcal A\) automatically inherit from the analytical object? The most satisfactory answers are given for the situation when \(\mathcal A\) is densely embedded into a reduced \(C^*\)-algebraic locally compact group \(A\) in such a way that the (left) Haar weight of \(A\) does not vanish on \(\mathcal A\). Then \(\mathcal A\) is shown to be an algebraic quantum group, and to consist of analytic elements for various automorphism groups naturally associated with \(A\). A corresponding result is proved for a regular multiplier Hopf \(^*\)-algebra. The authors present also a new, purely algebraic proof that each \(^*\)-algebraic quantum group possesses a right-invariant positive functional and discuss several examples of the embeddings being the subject of their study.
The first part of the paper contains a useful introduction to the studied concepts.

MSC:
16T05 Hopf algebras and their applications
46L65 Quantizations, deformations for selfadjoint operator algebras
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
22D35 Duality theorems for locally compact groups
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
16T20 Ring-theoretic aspects of quantum groups
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