The Haagerup property for locally compact quantum groups

Daws, M., Fima, P., Skalski, A. and White, S. (2016) The Haagerup property for locally compact quantum groups. Journal für die Reine und Angewandte Mathematik (Crelles Journal), 2016(711), pp. 189-229. (doi:10.1515/crelle-2013-0113)

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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 b 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.

Item Type:Articles
Additional Information:MD is partially supported by EPSRC grant EP/IO26819/1. PF is partially supported by the ANR grants NEUMANN and OSQPI. AS is partially supported by the Iuventus Plus grant IP2012 043872. SW is partially supported by EPSRC grant EP/IO19227/1-2.
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:White, Professor Stuart
Authors: Daws, M., Fima, P., Skalski, A., and White, S.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Journal für die Reine und Angewandte Mathematik (Crelles Journal)
Publisher:Walter de Gruyter
ISSN:0075-4102
ISSN (Online):1435-5345
Copyright Holders:Copyright © 2014 The Authors
First Published:First published in Journal für die Reine und Angewandte Mathematik (Crelles Journal) 2016(711):189-229
Publisher Policy:Reproduced under a Creative Commons License

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