On Amenability and Co-Amenability of Algebraic Quantum Groups and Their Corepresentations

Bédos, Erik; Conti, Roberto; Tuset, Lars

doi:10.4153/cjm-2005-002-8

Cited by 37 publications

(45 citation statements)

References 28 publications

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“…Similar considerations can be found in the articles [3,54]. Weak containment is defined in terms of the corresponding concept for representations of C -algebras, and so we begin by recalling this definition from [28,Section 3.4].…”

Section: Containment and Weak Containment Of Representations Of Localmentioning

confidence: 99%

“…On the other hand, if G is compact, then each representation which has almost invariant vectors actually has invariant vectors, so G has property (T) (see [3,Theorem 7.16…”

Section: The Haagerup Approximation Propertymentioning

confidence: 99%

“…ı U ; this is condition (iii) in the following proposition (for an alternative approach, see [5,Theorem 5.1]). In [3] and [5, Section 5] the terminology U has WCP (the weak containment property) is used for those representations U with O u 4 U ; here we use the terminology that U has almost invariant vectors for this condition.…”

Section: Containment and Weak Containment Of Representations Of Localmentioning

confidence: 99%

See 2 more Smart Citations

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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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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Section: Containment and Weak Containment Of Representations Of Localmentioning

confidence: 99%

“…On the other hand, if G is compact, then each representation which has almost invariant vectors actually has invariant vectors, so G has property (T) (see [3,Theorem 7.16…”

Section: The Haagerup Approximation Propertymentioning

confidence: 99%

Section: Containment and Weak Containment Of Representations Of Localmentioning

confidence: 99%

See 1 more Smart Citation

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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show abstract

“…The study of property (T) for quantum groups began before the paper [13]. In [27] property (T) was studied in the setting of Kac algebras and in [3] it was introduced for the class of algebraic quantum groups. As is shown in [23], these different notions all agree with Fima's definition in the case of a discrete quantum group.…”

Section: Preliminaries On Quantum Groupsmentioning

confidence: 99%

A cohomological description of property (T) for quantum groups

Kyed

2011

Journal of Functional Analysis

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We prove a Delorme-Guichardet type theorem for discrete quantum groups expressing property (T) of the quantum group in question in terms of its first cohomology groups. As an application, we show that the first L 2 -Betti number of a discrete property (T) quantum group vanishes.The notion of property (T) was introduced by Kazhdan in his influential paper [18] and has since then played a prominent role in a variety of mathematical disciplines, including topology, ergodic theory and operator algebras. Over the years the definition has been generalized to different operator algebraic settings, for instance by Connes and Jones for the class of II 1 -factors in [9] and by Bekka for tracial C * -algebras in [5]. Recently Fima introduced property (T) in the context of discrete quantum groups; a class of operator algebras not necessarily arising from groups, but still carrying some of the extra structure present in group C * -algebras or group von Neumann algebras. The present paper is devoted to the study of this notion of property (T). Before stating our main results, we set the stage by briefly discussing the definitions and a few classical results ✩

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“…We only treat a co-amenable G in this paper, and ε extends to the character on C(G). See [3][4][5]43] for details of the amenability.…”

Section: Compact Quantum Groupmentioning

confidence: 99%

Product type actions ofGq

Tomatsu

2015

Advances in Mathematics

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MSC: 46L55 20G42 17B37Keywords: Infinite tensor product type actions Depth 2 subfactors Non-commutative Poisson boundaries Quantum flag manifoldsWe will study a faithful product type action of G q , the q-deformation of a connected semisimple compact Lie group G, and prove that such an action is induced from a minimal action of the maximal torus T of G q . This enables us to classify product type actions of SU q (2) up to conjugacy. We also compute the intrinsic group of G q,Ω , the 2-cocycle deformation of G q that is naturally associated with the quantum flag manifold L ∞ (T \G q ).

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On Amenability and Co-Amenability of Algebraic Quantum Groups and Their Corepresentations

Cited by 37 publications

References 28 publications

The Haagerup property for locally compact quantum groups

The Haagerup property for locally compact quantum groups

A cohomological description of property (T) for quantum groups

Product type actions ofGq

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