Amenability and Co-Amenability for Locally Compact Quantum Groups

Bédos, Erik; Tuset, Lars

doi:10.1142/s0129167x03002046

Cited by 110 publications

(124 citation statements)

References 14 publications

Supporting

Mentioning

124

Contrasting

Order By: Relevance

“…The equivalences (i) , (ii) , (iii) , (iv) , (vi) are either straightforward or can be viewed as variants of the results obtained in the articles [4,5]. We thus look only at (v).…”

Section: Containment and Weak Containment Of Representations Of Localmentioning

confidence: 96%

“…As the theory reached a certain level of maturity, it became natural to investigate questions relating quantum groups to noncommutative probability, noncommutative geometry, and analyse actions of quantum groups. In particular a study of approximationtype/geometric properties such as amenability (see [5] and references there) or property (T) [31,54] has recently been initiated, often with a special focus on the case of discrete quantum groups. Recently Brannan established the Haagerup property for the von Neumann algebras associated to free orthogonal and unitary quantum groups [14] (analogous results have also been obtained for the von Neumann algebras associated to certain quantum automorphism groups [15], and quantum reflection groups [56]).…”

Section: Introductionmentioning

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)

View full text Add to dashboard Cite

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.

show abstract

“…The equivalences (i) , (ii) , (iii) , (iv) , (vi) are either straightforward or can be viewed as variants of the results obtained in the articles [4,5]. We thus look only at (v).…”

Section: Containment and Weak Containment Of Representations Of Localmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

The Haagerup property for locally compact quantum groups

Daws

Fima

Skalski

et al. 2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

show abstract

“…Analogously, these hold for the right module action. Suppose now that we have some left-module action of M (A) on M (E) satisfying (1) and (2)…”

Section: For Banach Algebrasmentioning

confidence: 99%

Multipliers, self-induced and dual Banach algebras

Daws¹

2010

Dissertationes Math.

View full text Add to dashboard Cite

In the first part of the paper, we present a short survey of the theory of multipliers, or double centralisers, of Banach algebras and completely contractive Banach algebras. Our approach is very algebraic: this is a deliberate attempt to separate essentially algebraic arguments from topological arguments. We concentrate upon the problem of how to extend module actions, and homomorphisms, from algebras to multiplier algebras. We then consider the special cases when we have a bounded approximate identity, and when our algebra is self-induced. In the second part of the paper, we mainly concentrate upon dual Banach algebras. We provide a simple criterion for when a multiplier algebra is a dual Banach algebra. This is applied to show that the multiplier algebra of the convolution algebra of a locally compact quantum group is always a dual Banach algebra. We also study this problem within the framework of abstract Pontryagin duality, and show that we construct the same weak * topology. We explore the notion of a Hopf convolution algebra, and show that in many cases, the use of the extended Haagerup tensor product can be replaced by a multiplier algebra.Acknowledgements. I wish to thank the referees for their careful and timely reading of this manuscript, and for comments which have improved the exposition.2010 Mathematics Subject Classification: Primary 43A20, 43A30, 46H05, 46H25, 46L07, 46L89; Secondary 16T05, 43A22, 81R50. Key words and phrases: multiplier, double centraliser, Fourier algebra, locally compact quantum group, dual Banach algebra, Hopf convolution algebra. Received 12.2.2010; revised version 29.4.2010.[4] IntroductionMultipliers are a useful way of embedding a non-unital algebra into a unital algebra: a problem which occurs often in algebraic analysis. The theory has reached maturity when applied to C * -algebras (see, for example, [59, Chapter 2]) where it is best studied in the context of Hilbert C * -modules, [33]. Indeed, one can also study "unbounded operators" for C * -algebras, [61], which form a vital tool in the study of quantum groups. For Banach algebras with a bounded approximate identity, much of the theory carries over (see [24, Section 1.d] or [8, Theorem 2.9.49]) although we remark that there seems to be no parallel to the unbounded theory.This paper starts with a survey of multipliers; we start with some generality, working with multipliers of modules, and not just algebras. This material is surely well-known to experts, but we are not aware of any particularly definitive source. For example, in [41], Ng uses similar ideas (but for C * -algebras, working in the category of operator modules) motived by the study of cohomology theories for Hopf operator algebras (that is, loosely speaking, quantum groups). However, most of the proofs are left in an unpublished manuscript. The particular aspects of the theory which we develop are somewhat motivated by Ng's presentation.We quickly turn to discussing Banach algebras, but we shall not (as is usually the case) require a bounded app...

show abstract

“…Examples of counital C * -bialgebras include all locally compact quantum groups in the universal setting [13] -in particular, the coamenable locally compact quantum groups [4].…”

Section: Multiplier C * -Bialgebrasmentioning

confidence: 99%

Quantum Random Walk Approximation on Locally Compact Quantum Groups

Lindsay

Skalski

2013

Lett Math Phys

View full text Add to dashboard Cite

Abstract.A natural scheme is established for the approximation of quantum Lévy processes on locally compact quantum groups by quantum random walks. We work in the somewhat broader context of discrete approximations of completely positive quantum stochastic convolution cocycles on C * -bialgebras. Mathematics Subject Classification (2000). Primary 46L53, 81S25; Secondary 22A30, 47L25, 16W30.Keywords. quantum random walk, quantum Lévy process, noncommutative probability, locally compact quantum group, C * -bialgebra, stochastic cocycle. IntroductionIn [19] we developed a theory of quantum stochastic convolution cocycles on counital multiplier C * -bialgebras, extending the algebraic theory of quantum Lévy processes created by Schürmann and co-workers (see [25] and references therein, and, for a simplified treatment [17]), and the topological theory of quantum stochastic convolution cocycles on compact quantum groups and operator space coalgebras developed by the authors [18]. Here we apply the results of [19] to introduce and analyse a straightforward scheme for the approximation of such cocycles by quantum random walks. In particular we obtain results on Markovregular quantum Lévy processes on locally compact quantum semigroups, extending and strengthening results in [11] for the compact case. Our analysis exploits a recent approximation theorem of Belton [6], which extends that of [24] (used in [11]). The approximation scheme closely mirrors the way in which Picard iteration operates in the construction of solutions of quantum stochastic differential equations [15].The study of quantum random walks on quantum groups was initiated by Biane in the early 1990s (starting with [7]). Some combinatorial, probabilistic and physical

show abstract

Amenability and Co-Amenability for Locally Compact Quantum Groups

Cited by 110 publications

References 14 publications

The Haagerup property for locally compact quantum groups

The Haagerup property for locally compact quantum groups

Multipliers, self-induced and dual Banach algebras

Quantum Random Walk Approximation on Locally Compact Quantum Groups

Contact Info

Product

Resources

About