We investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group. Two definitions -one due to S. Vaes and one due to S.L. Woronowicz -are analyzed and relations between them discussed. Among many reformulations we prove that the former definition can be phrased in terms of quasi-equivalence of representations of quantum groups while the latter can be related to an old definition of Podleś from the theory of compact quantum groups. The cases of classical groups, duals of classical groups, compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context. A deep relationship with the quantum group generalization of Herz restriction theorem from classical harmonic analysis is also established, in particular, in the course of our analysis we give a new proof of Herz restriction theorem.

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.

We study representations of Banach algebras on reflexive Banach spaces. Algebras which admit such representations which are bounded below seem to be a good generalisation of Arens regular Banach algebras; this class includes dual Banach algebras as defined by Runde, but also all group algebras, and all discrete (weakly cancellative) semigroup algebras. Such algebras also behave in a similar way to C *and W * -algebras; we show that interpolation space techniques can be used in the place of GNS type arguments. We define a notion of injectivity for dual Banach algebras, and show that this is equivalent to Connes-amenability. We conclude by looking at the problem of defining a well-behaved tensor product for dual Banach algebras.

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...

Abstract:We study Property (T) for locally compact quantum groups, providing several new characterisations, especially related to operator algebraic ergodic theory. Quantum Property (T) is described in terms of the existence of various Kazhdan type pairs, and some earlier structural results of Kyed, Chen and Ng are strengthened and generalised. For second countable discrete unimodular quantum groups with low duals, Property (T) is shown to be equivalent to Property (T) 1,1 of Bekka and Valette. This is used to extend to this class of quantum groups classical theorems on 'typical' representations (due to Kerr and Pichot), and on connections of Property (T) with spectral gaps (due to Li and Ng) and with strong ergodicity of weakly mixing actions on a particular von Neumann algebra (due to Connes and Weiss). Finally, we discuss in the Appendix equivalent characterisations of the notion of a quantum group morphism with dense image.