# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] S. Baaj and S. Vaes, Double crossed products of locally compact quantum groups , J. Institute Math. Jussieu 4 (2005), 135-173. · Zbl 1071.46040 · doi:10.1017/S1474748005000034 [2] E. Bédos, G.J. Murphy and L. Tuset, Co-amenability of compact quantum groups , J. Geom. Phys. 40 (2001), 130-153. · Zbl 1011.46056 · doi:10.1016/S0393-0440(01)00024-9 [3] L. Delvaux and A. Van Daele, Algebraic quantum hypergroups , Adv. Math., · Zbl 1206.43004 [4] B. Drabant, A. Van Daele and Y. Zhang, Actions of multiplier Hopf algebras , Comm. Algebra 27 (1999), 4117-4172. · Zbl 0951.16013 · doi:10.1080/00927879908826688 [5] A. Klimyk and K. Schmudgen, Quantum groups and their representations , Springer, Berlin, 1997. [6] A.W. Knapp, Lie groups, Lie algebras and cohomology , Princeton Univ. Press, Princeton, NJ, 1988. · Zbl 0648.22010 [7] J. Kustermans, KMS- weights on C $$^*$$-algebras, Preprint Odense Universitet (1997), · arxiv:math.funct-an/9704008. [8] ——–, One-parameter representations on C $$^*$$-algebras, Preprint Odense Universitet (1997), · arxiv:math.funct-an/9707009. [9] ——–, The analytic structure of an algebraic quantum group , J. Algebra 259 (2003), 415-450. · Zbl 1034.46064 · doi:10.1016/S0021-8693(02)00570-7 [10] J. Kustermans and S. Vaes, Weight theory for $$C^*$$-algebraic quantum groups , Preprint KU Leuven and University College Cork (1999), · Zbl 0957.46037 · doi:10.1016/S0764-4442(99)80288-2 · arxiv:math.OA/9901063. [11] ——–, Locally compact quantum groups , Ann. Sci. Ecol. Norm. Sup. 33 (2000), 837-934. · Zbl 1034.46508 · doi:10.1016/S0012-9593(00)01055-7 · numdam:ASENS_2000_4_33_6_837_0 · eudml:82536 [12] ——–, Locally compact quantum groups in the von Neumann algebra setting , Math. Scand. 92 (2003), 68-92. · Zbl 1034.46067 [13] J. Kustermans and A. Van Daele, C*- algebraic quantum groups arising from algebraic quantum groups , Inter. Jour. Math. 8 (1997), 1067-1139. · Zbl 1009.46038 · doi:10.1142/S0129167X97000500 [14] M.B. Landstad and A. Van Daele, Groups with compact open subgroups and multiplier Hopf $$^*$$-algebras , Expo. Math. 26 (2008), 197-217. · Zbl 1149.22005 · doi:10.1016/j.exmath.2007.10.004 [15] ——–, Compact and discrete subgroups of algebraic quantum groups , K.U. Leuven and University of Trondheim, [16] T. Masuda, Y. Nakagami and S.L. Woronowicz, A C $$^*$$-algebraic framework for quantum groups, Inter. J. Math. 14 (2003), 903-1001. · Zbl 1053.46050 · doi:10.1142/S0129167X03002071 [17] M. Takesaki, Theory of operator algebras II, Springer, Berlin, 2003. · Zbl 1059.46031 [18] S. Vaes, A Radon-Nikodym theorem for von Neumann algebras , J. Operator Theory 46 (2001), 477-489. · Zbl 0995.46042 [19] S. Vaes and A. Van Daele, The Heisenberg commutation relations, commuting squares and the Haar measure on locally compact quantum groups , in Operator algebras and mathematical physics : Conference proceedings , Constanta (Romania), 2001. · Zbl 1247.46061 [20] A. Van Daele, Multiplier Hopf algebras , Trans. Amer. Math. Soc. 342 (1994), 917-932. · Zbl 0809.16047 · doi:10.2307/2154659 [21] ——–, An algebraic framework for group duality , Adv. Math. 140 (1998), 323-366. · Zbl 0933.16043 · doi:10.1006/aima.1998.1775 [22] ——–, Quantum groups with invariant integrals , PNAS 97 (2000), 541-556. · Zbl 0984.16038 · doi:10.1073/pnas.97.2.541 [23] ——–, The Haar measure on some locally compact quantum groups , · Zbl 1064.46059 · arxiv:math.OA [24] ——–, Locally compact quantum groups. A von Neumann algebra approach , · Zbl 1034.46508 · arxiv:math.OA [25] ——–, Discrete quantum groups , J. Algebra 180 (1994), 431-444. · Zbl 0864.17012 · doi:10.1006/jabr.1996.0075 [26] A. Van Daele and Y. Zhang, A survey on multiplier Hopf algebras , · Zbl 1020.16032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.