International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 10, Pages 577-601

Amenability and coamenability of algebraic quantum groups

Erik Bédos,1 Gerard J. Murphy,2 and Lars Tuset3

1Institute of Mathematics, University of Oslo, P.B. 1053 Blindern, Oslo 0316, Norway
2Department of Mathematics, National University of Ireland, Cork, Ireland
3Faculty of Engineering, Oslo University College, Cort Adelers Gate 30, Oslo 0254, Norway

Received 5 June 2001; Revised 25 January 2002

Copyright © 2002 Erik Bédos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We define concepts of amenability and coamenability for algebraic quantum groups in the sense of Van Daele (1998). We show that coamenability of an algebraic quantum group always implies amenability of its dual. Various necessary and/or sufficient conditions for amenability or coamenability are obtained. Coamenability is shown to have interesting consequences for the modular theory in the case that the algebraic quantum group is of compact type.