Illinois Journal of Mathematics

On a quantum version of Ellis joint continuity theorem

Biswarup Das and Colin Mrozinski

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


We give a necessary and sufficient condition on a compact semitopological quantum semigroup which turns it into a compact quantum group. We give two applications of our results: a “noncommutative” version of Ellis joint continuity theorem for semitopological groups, a corollary to which is a new C∗-algebraic proof of the theorem for classical semitopological semigroup; we also investigate the question of the existence of the Haar state on a compact semitopological quantum semigroup and prove a “noncommutative” version of the converse Haar’s theorem.

Article information

Illinois J. Math., Volume 59, Number 4 (2015), 839-858.

Received: 14 August 2015
Revised: 18 July 2016
First available in Project Euclid: 27 February 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 81R15: Operator algebra methods [See also 46Lxx, 81T05] 22A15: Structure of topological semigroups 22A20: Analysis on topological semigroups 42B35: Function spaces arising in harmonic analysis 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37]


Das, Biswarup; Mrozinski, Colin. On a quantum version of Ellis joint continuity theorem. Illinois J. Math. 59 (2015), no. 4, 839--858. doi:10.1215/ijm/1488186012.

Export citation


  • E. Bédos, G. J. Murphy and L. Tuset, Co-amenability of compact quantum groups, J. Geom. Phys. 40 (2001), 130–153.
  • D. P. Blecher and R. R. Smith, The dual of the Haagerup tensor product, J. Lond. Math. Soc. (2) 45 (1992), 126–144.
  • R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach Science Publishers, New York-London-Paris, 1970.
  • A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I. Mathematical Surveys, vol. 7. American Math. Society, Providence, 1961.
  • B. Das and M. Daws, Quantum Eberlein compactifications and invariant means, Indiana Univ. Math. J. 65 (2016), 307–352.
  • M. Daws, Characterising weakly almost periodic functionals on the measure algebra. Studia Math. 204 (2011), 213–234.
  • M. Daws, Noncommutative separate continuity and weakly almost periodicity for Hopf von Neumann algebras. J. Funct. Anal. 269 (2015), no. 3, 683–704.
  • K. De Leeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math. 105 (1961), 63–97.
  • R. Ellis, Locally compact transformation groups, Duke Math. J. 24 (1957), no. 2, 119–125.
  • M. Enock and J. M. Schwartz, Kac algebras and duality of locally compact groups, Springer-Verlag, Berlin, 1992.
  • J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. Éc. Norm. Supér. 33 (2000), no. 6, 837–934.
  • A. Maes and A. Van Daele, Notes on compact quantum groups, Nieuw Arch. Wiskd. (4) 16 (1998), 73–112.
  • M. G. Megrelishvili, V. G. Pestov and V. V. Uspenskij A note on the precompactness of weakly almost periodic groups, Res. Exp. Math. 24 (2001), 209–216.
  • A. Mukherjea and N. A. Tserpes, Invariant measures and the converse of Haar's theorem on semitopological semigroups, Pacific J. Math. 44 (1973), no. 1, 251–262.
  • G. J. Murphy and L. Tuset, Aspects of compact quantum group theory, Proc. Amer. Math. Soc. 10 (2004), no. 132, 3055–3067.
  • P. Sołtan, Quantum Bohr compactification, Illinois J. Math. 49 (2005), no. 4, 1245–1270.
  • S. L. Woronowicz, Compact quantum groups, Symétries quantiques, pp. 845–884, 1998.