Amenable discrete quantum groups

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 O 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.

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“…This follows from Proposition 6.2 and the fact that amenability of G implies the coamenability of O G, shown in [70].…”

Section: Haagerup Approximation Property For Discrete Quantum Groupsmentioning

confidence: 75%

The Haagerup property for locally compact quantum groups

Daws

Fima

Skalski

et al. 2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…In the classical case, the following theorem is proved in [Eff75]. The quantum case is more involved and we use some techniques from [Tom06].…”

Section: As a Resultsmentioning

confidence: 99%

A Cocycle in the Adjoint Representation of the Orthogonal Free Quantum Groups

Fima

Vergnioux

2015

Int Math Res Notices

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Abstract. We show that the orthogonal free quantum groups are not inner amenable and we construct an explicit proper cocycle weakly contained in the regular representation. This strengthens the result of Vaes and the second author, showing that the associated von Neumann algebras are full II 1 -factors and Brannan's result showing that the orthogonal free quantum groups have Haagerup's approximation property. We also deduce Ozawa-Popa's property strong (HH) and give a new proof of Isono's result about strong solidity.

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“…The result follows by a standard argument similar to the Namioka's idea in [11](see also [19,Lemma 3.5]); however, we give a sketch of the proof.…”

Section: Amenability Of Actions Of Lau Algebrasmentioning

confidence: 60%

“…It is known that, if G is co-amenable, then G is amenable [1,Theorem 3.2]. The converse implication is known to hold in the discrete case [19].…”

Section: Introductionmentioning

confidence: 99%

On Action of Lau Algebras on Von Neumann Algebras

Ramezanpour¹

2015

Bulletin of the Korean Mathematical Society

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Abstract. Let G be a von Neumann algebraic locally compact quantum group, in the sense of Kustermans and Vaes. In this paper, as a consequence of a notion of amenability for actions of Lau algebras, we show that G, the dual of G, is co-amenable if and only if there is a state m ∈ L ∞ ( G) * which is invariant under a left module action of L 1 (G) on L ∞ ( G) * . This is the quantum group version of a result by Stokke [17]. We also characterize amenable action of Lau algebras by several properties such as fixed point property. This yields in particular, a fixed point characterization of amenable groups and H-amenable representation of groups.

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Amenable discrete quantum groups

Cited by 68 publications

References 14 publications

The Haagerup property for locally compact quantum groups

The Haagerup property for locally compact quantum groups

A Cocycle in the Adjoint Representation of the Orthogonal Free Quantum Groups

On Action of Lau Algebras on Von Neumann Algebras

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