2013/03/31 by Matthew Daws, Pierre Fima, Adam Skalski +1 · 1 citation
Mathematics · #math.OA #math.QA
paper · pdf · doi:10.1515/crelle-2013-0113
published as J. Reine Angew. Math. 711 (2016), 189-229 · 37 pages; v3 simplifies and shortens some arguments (mainly in Sections 1,2 and 6), corrects a few minor points and adds further references. The paper will appear in Journal für die reine und angewandte Mathematik
arxiv created 2013/11/11 · arxiv updated 2016/02/16
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.