Annals of Functional Analysis

Bekka-type amenabilities for unitary corepresentations of locally compact quantum groups

Xiao Chen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this short note, we further Ng’s work by extending Bekka amenability and weak Bekka amenability to general locally compact quantum groups, and we generalize some of Ng’s results to the general case. In particular, we show that a locally compact quantum group G is coamenable if and only if the contra-corepresentation of its fundamental multiplicative unitary WG is Bekka-amenable, and that G is amenable if and only if its dual quantum group’s fundamental multiplicative unitary WGˆ is weakly Bekka-amenable.

Article information

Ann. Funct. Anal., Volume 9, Number 2 (2018), 210-219.

Received: 23 January 2017
Accepted: 5 May 2017
First available in Project Euclid: 7 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20G42: Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50]
Secondary: 46L89: Other "noncommutative" mathematics based on C-algebra theory [See also 58B32, 58B34, 58J22] 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx]

locally compact quantum group Bekka amenability weak Bekka amenability coamenability amenability


Chen, Xiao. Bekka-type amenabilities for unitary corepresentations of locally compact quantum groups. Ann. Funct. Anal. 9 (2018), no. 2, 210--219. doi:10.1215/20088752-2017-0044.

Export citation


  • [1] S. Baaj and G. Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de $C^{*}$-algèbres, Ann. Sci. Éc. Norm. Supér. (4) 26 (1993), no. 4, 425–488.
  • [2] E. Bédos, R. Conti, and L. Tuset, On amenability and co-amenability of algebraic quantum groups and their corepresentations, Canad. J. Math. 57 (2005), no. 1, 17–60.
  • [3] E. Bédos and L. Tuset, Amenability and co-amenability for locally compact quantum groups, Internat. J. Math. 14 (2003), no. 8, 865–884.
  • [4] M. E. B. Bekka, Amenable unitary representations of locally compact groups, Invent. Math. 100 (1990), no. 2, 383–401.
  • [5] M. Enock and J.-M. Schwartz, Algébres de Kac moyennables, Pacific J. Math. 125 (1986), no. 2, 363–379.
  • [6] J. Kustermans, Locally compact quantum groups in the universal setting, Internat. J. Math. 12 (2001), no. 3, 289–338.
  • [7] J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. Éc. Norm. Supér. (4) 33 (2000), no. 6, 837–934.
  • [8] J. Kustermans and S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (2003), no. 1, 68–92.
  • [9] C.-K. Ng, “Amenability of Hopf $C^{*}$-algebras” in Operator Theoretical Methods (Timişoara, 1998), Theta Found., Bucharest, 2000, 269–284.
  • [10] C.-K. Ng, Amenable representations and Reiter’s property for Kac algebras, J. Funct. Anal. 187 (2001), no. 1, 163–182.
  • [11] C.-K. Ng, An example of amenable Kac systems, Proc. Amer. Math. Soc. 130 (2002), no. 10, 2995–2998.
  • [12] J.-P. Pier, Amenable Locally Compact Groups, Pure Appl. Math., Wiley, New York, 1984.
  • [13] Z.-J. Ruan, Amenability of Hopf von Neumann algebras and Kac algebras, J. Funct. Anal. 139 (1996), no. 2, 466–499.
  • [14] V. Runde, Lectures on Amenability, Lecture Notes in Math. 1774, Springer, Berlin, 2002.
  • [15] T. Timmermann, An Invitation to Quantum Groups and Duality: From Hopf Algebras to Multiplicative Unitaries and Beyond, EMS Textbk. Math., Eur. Math. Soc., Zürich, 2008.
  • [16] D. Voiculescu, “Amenability and Katz algebras” in Algèbres d’opérateurs et leurs applications en physique mathématique (Marseille, 1977), Colloq. Internat. CNRS N274, CNRS, Paris, 1979, 451–457.