The Haagerup property for locally compact quantum groups
The Haagerup property for locally compact quantum groups
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DOI：
10.1515/crelle-2013-0113
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发表时间：
2013-03
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影响因子：
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通讯作者：
Matthew Daws;Pierre Fima;Adam G. Skalski;Stuart White
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文献类型：
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作者：
Matthew Daws;Pierre Fima;Adam G. Skalski;Stuart White
将局部紧群的Haagerup性质推广到局部紧量子群的情形，并利用酉表示和正定函数建立了几个等价的刻画.特别地，证明了局部紧量子群G具有Haagerup性质当且仅当其混合表示在所有么正表示的空间中稠密。对于离散的G，我们通过对偶量子群$\hat{G}$上的对称真条件负泛函的存在性证明了Haagerup性质，通过G上的真实的真上圈的存在性证明了Haagerup性质，进一步，如果G也是幺模的，我们证明了Haagerup性质是G的von Neumann性质.这将Akemann，Walter，Bekka，Cherix，Valette和Jolissaint的结果扩展到量子环境，并提供了与Brannan最近工作的联系。我们使用这些特征表明，Haagerup属性下的离散量子群的自由产品保持。
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 $\hat{G}$; 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.