Abstract
The Haagerup property for locally compact groups is generalised to the context of locally compact quantum groups, with several equivalent characterisations in terms of the unitary representations and positive-definite functions established. In particular it is shown that a locally compact quantum group 𝔾 has the Haagerup property if and only if its mixing representations are dense in the space of all unitary representations. For discrete 𝔾 we characterise the Haagerup property by the existence of a symmetric proper conditionally negative functional on the dual quantum group
Funding source: EPSRC
Award Identifier / Grant number: EP/IO26819/1
Funding source: ANR
Award Identifier / Grant number: NEUMANN, OSQPI
Funding source: Iuventus Plus
Award Identifier / Grant number: IP2012 043872
Funding source: EPSRC
Award Identifier / Grant number: EP/IO19227/1-2, EP/I026819/I
Some work on this paper was undertaken during a visit of AS and SW to the University of Leeds in June 2012, funded by EPSRC grant EP/I026819/I. They thank the faculty of the School of Mathematics for their hospitality. The authors would also like to thank Jan Cameron, Caleb Eckhardt, David Kyed, Roland Vergnioux and the anonymous referee for valuable comments and advice.
© 2016 by De Gruyter
This article is distributed under the terms of the Creative Commons Attribution 3.0 Public License.