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On Amenability and Co-Amenability of Algebraic Quantum Groups and Their Corepresentations

Published online by Cambridge University Press:  20 November 2018

Erik Bédos
Institute of Mathematics, University of Oslo, P.B. 1053 Blindern, 0316 Oslo, Norway e-mail:
Roberto Conti
Mathematisches Institut, Friedrich-Alexander Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2 D-91054 Erlangen, Germany e-mail:
Lars Tuset
Faculty of Engineering, Oslo University College, Cort Adelers Gate 30, 0254 Oslo, Norway e-mail:
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We introduce and study several notions of amenability for unitary corepresentations and $*$-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background for this study, we investigate the associated tensor ${{C}^{*}}$-categories.

Research Article
Copyright © Canadian Mathematical Society 2005


[1] Baaj, S. and Skandalis, G., Unitaires multiplicatifs et dualité pour les produits croisés de C*- algèbres. Ann. Sci. École. Norm. Sup. 26 (1993), 425488.Google Scholar
[2] Banica, T., Representations of compact quantum groups and subfactors. J. Reine Angew.Math. 509 (1999), 167198.Google Scholar
[3] Banica, T., Fusion rules for representations of compact quantum groups. Exposition. Math. 17 (1999), 313338.Google Scholar
[4] Bédos, E., Notes on hypertraces and C*-algebras. J. Operator Theory 34 (1995), 285306.Google Scholar
[5] Bédos, E., Murphy, G. J. and Tuset, L., Co-amenability of compact quantum groups. J. Geom. Phys. 40 (2001), 130153.Google Scholar
[6] Bédos, E., Murphy, G. J. and Tuset, L., Amenability and co-amenability for algebraic quantum groups. Int. J. Math. Math. Sci. 31 (2002), 577601.Google Scholar
[7] Bédos, E., Murphy, G. J. and Tuset, L., Amenability and co-amenability for algebraic quantum groups, II. J. Funct. Anal. 201 (2003), 303340.Google Scholar
[8] Bekka, M. E. B., Amenable unitary representation of locally compact groups. Invent.Math. 100 (1990), 383401 .Google Scholar
[9] Blanchard, E., Déformations de C*- Algèbres de Hopf. Bull. Soc. Math. France 124 (1996), 141215.Google Scholar
[10] Blanchard, E. and Vaes, S., A remark on amenability of discrete quantum groups. Preliminary version (2002).Google Scholar
[11] Desmedt, P., Quaegebeur, J. and Vaes, S., Amenability and the bicrossed product construction. Illinois J. Math. 46 (2003), 12591273.Google Scholar
[12] Dixmier, J., Les C*-algèbres et leurs représentations. Gauthiers-Villars, Paris, 1969.Google Scholar
[13] Effros, E. G. and Ruan, Z.-J., Discrete quantum groups. I. The Haar measure. Internat. J. Math. 5 (1994), 681723.Google Scholar
[14] Enock, M. and Schwartz, J. M., Kac algebras and duality of locally compact groups. Springer-Verlag, Berlin, 1992.Google Scholar
[15] Enock, M. and Schwartz, J. M., Algèbres de Kac moyennables. Pacific. J. Math. 125 (1986), 363379.Google Scholar
[16] Fell, J. M. G., Weak containment and induced representations of groups. Canad. J. Math. 14 (1962), 237268.Google Scholar
[17] Ghez, L. R. and Roberts, J. E.,W*-categories. Pacific. J. Math. 120 (1985), 79109.Google Scholar
[18] Greenleaf, F. P., Invariant means on topological groups. Van Nostrand, New York, 1969.Google Scholar
[19] de la Harpe, P., and Valette, A., La propriété. (T) de Kazhdan pour les groupes localement compacts. Asterisque 175, Soc. Math. de France, 1989.Google Scholar
[20] Jacobs, A., and Van Daele, A., The quantum E(2) as a locally compact quantum group. Preprint, K.U. Leuven (2003) (in preparation).Google Scholar
[21] Kustermans, J., Examining the dual of an algebraic quantum group. Preprint Odense Universitet (1997). (arXiv:funct-an/9704006).Google Scholar
[22] Kustermans, J., Universal C*- algebraic quantum groups arising from algebraic quantum groups. Preprint Odense Universitet (1997). (arXiv:funct-an/9704004).Google Scholar
[23] Kustermans, J. and Van Daele, A., C*-algebraic quantum groups arising from algebraic quantum groups. Int. J. Math. 8 (1997), 10671139.Google Scholar
[24] Kustermans, J., and Vaes, S., Locally compact quantum qroups. Ann. Sci. École Norm. Sup. 33 (2000), 837934.Google Scholar
[25] Longo, R., and Roberts, J. E., A theory of dimension. K–Theory 11 (1997), 103159.Google Scholar
[26] Ng, C.-K., Amenability of Hopf C*-algebras. In: Operator theoretical works. Theta Found., Bucharest, 2000, pp. 269284.Google Scholar
[27] Ng, C.-K., An Example of amenable Kac algebras. Proc. Amer.Math. Soc. 130 (2002), 29952998.Google Scholar
[28] Ng, C.-K., Amenable representations and Reiter's property for Kac algebras. J. Funct. Anal. 187 (2001), 163182.Google Scholar
[29] Paterson, A. L., Amenability. Math. Surveys and Monographs 29, American Mathematical Society, Providence, RI, 1988.Google Scholar
[30] Petrescu, S. and Joita, M., Propert. (T) for Kac algebras. Rev. Roum. Math. Pures Appl. 37 (1992), 163178.Google Scholar
[31] Podles, P. and Woronowicz, S. L., Quantum deformation of Lorentz group. Comm. Math. Phys. 130 (1990), 381431.Google Scholar
[32] Roberts, J. E., and Tuset, L., On the equality of q-dimension and intrinsic dimension. J. Pure Appl. Algebra 156 (2001), 329343 .Google Scholar
[33] Ruan, Z.-J., Amenability of Hopf von Neumann algebras and Kac algebras. J. Funct. Anal. 139 (1996), 466499.Google Scholar
[34] Stratila, S., Modular theory in operator algebras. Abacus Press, TunbridgeWells, Kent, 1981.Google Scholar
[35] Tomatsu, R., Amenable discrete quantum groups. Preprint, University of Tokyo (2003).Google Scholar
[36] Van Daele, A., Multiplier Hopf algebras. Trans. Amer.Math. Soc. 342 (1994), 917932.Google Scholar
[37] Van Daele, A., An algebraic framework for group duality. Adv.Math. 140 (1998), 323366.Google Scholar
[38] Voiculescu, D., Amenability and Katz Algebras. Algèbres d’Opérateurs et leurs Applications en Physique Mathématique, 274, Colloq. Internation. C.N.R.S., 1977, pp. 451457.Google Scholar
[39] Woronowicz, S. L., Compact matrix pseudogroups. Comm. Math. Phys. 111 (1987), 613665.Google Scholar
[40] Woronowicz, S. L., Compact quantum groups. In: Symétries Quantiques, North Holland, Amsterdam, 1998, pp. 845884.Google Scholar