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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

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Volume 2016, Issue 711


The Haagerup property for locally compact quantum groups

Matthew Daws / Pierre Fima
  • Institut Mathématiques de Jussieu, Université Denis-Diderot Paris 7, 175, rue du Chevaleret, 75013 Paris, France
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/ Adam Skalski
  • Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00–956 Warsaw; and Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
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/ Stuart White
Published Online: 2014-01-10 | DOI: https://doi.org/10.1515/crelle-2013-0113


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 𝔾^; by the existence of a real proper cocycle on 𝔾, and further, if 𝔾 is also unimodular we show that the Haagerup property is a von Neumann property of 𝔾. This extends results of Akemann, Walter, Bekka, Cherix, Valette, and Jolissaint to the quantum setting and provides a connection to the recent work of Brannan. We use these characterisations to show that the Haagerup property is preserved under free products of discrete quantum groups.

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.

About the article

Received: 2013-05-22

Revised: 2013-11-06

Published Online: 2014-01-10

Published in Print: 2016-02-01

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

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2016, Issue 711, Pages 189–229, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2013-0113.

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© 2016 by De Gruyter. This article is distributed under the terms of the Creative Commons Attribution 3.0 Public License. BY 3.0

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