Projectivity of modules over Fourier algebras

Forrest, Brian; Lee, Hun Hee; Samei, Ebrahim

doi:10.1112/plms/pdq030

Cited by 13 publications

(16 citation statements)

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“…An exception is Proposition 5.3, which we check does translate; this was shown in [16,Proposition 3.3], but in the interests of completeness, we give a proof here (which we think is shorter).…”

Section: Completely Contractive Banach Algebrasmentioning

confidence: 99%

Multipliers, self-induced and dual Banach algebras

Daws¹

2010

Dissertationes Math.

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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...

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Section: Completely Contractive Banach Algebrasmentioning

confidence: 99%

Multipliers, self-induced and dual Banach algebras

Daws¹

2010

Dissertationes Math.

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“…The aim is to characterize homological properties of the Banach algebra 1 (S) (and its modules) in terms of the underlying semigroup S. Homological properties of Banach algebras associated with groups and semigroups have been studied by many authors. Some recent papers are [1,[6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Homological properties of modules over semigroup algebras

Ramsden

2010

Journal of Functional Analysis

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Let S be a semigroup. In this paper we investigate the injectivity of 1 (S) as a Banach right module over 1 (S). For weakly cancellative S this is the same as studying the flatness of the predual left module c 0 (S). For such semigroups S, we also investigate the projectivity of c 0 (S). We prove that for many semigroups S for which the Banach algebra 1 (S) is non-amenable, the 1 (S)-module 1 (S) is not injective. The main result about the projectivity of c 0 (S) states that for a weakly cancellative inverse semigroup S, c 0 (S) is projective if and only if S is finite.

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“…The present paper is related to a collection of papers studying module structures of L pspaces associated to the Fourier algebra of locally compact groups [3], [4], [5]. These papers are based on the theory of non-commutative L p -spaces associated to arbitrary (not necessarily semi-finite) von Neumann algebras, which we recall below.…”

Section: It Is Known That the Fourier Transform Can Be Generalized Inmentioning

confidence: 99%

“…In fact, there is a choice which determines the intersection and which depends on a complex interpolation parameter z ∈ C. In [5] the parameter z = −1/2 is used, whereas [3] focuses on the case z = 0 in order to define module actions. In the final remarks of [4], it is questioned which parameter would fit best for quantum groups.…”

Section: It Is Known That the Fourier Transform Can Be Generalized Inmentioning

confidence: 99%

“…In particular, they mention that it remains unclear how the L p -space associated with a quantum group should be turned into a L 1 -module. In [5] this is done using Izumi's L p -spaces for the complex interpolation parameter z = −1/2, whereas in [3] a similar, but not identical construction was used for the parameter z = 0. We believe that the Fourier transform indicates that the most natural choice would be z = −1/2.…”

Section: A Distinguished Choice For the Interpolation Parametermentioning

confidence: 99%

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The $L^p$-Fourier transform on locally compact quantum groups

Caspers¹

2013

J. Operator Theory

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Abstract. Using interpolation properties of non-commutative L p -spaces associated with an arbitrary von Neumann algebra, we define a L p -Fourier transform 1 ≤ p ≤ 2 on locally compact quantum groups. We show that the Fourier transform determines a distinguished choice for the interpolation parameter as introduced by Izumi. We define a convolution product in the L p -setting and show that the Fourier transform turns the convolution product into a product. IntroductionThe Fourier transform is one of the most powerful tools coming from abstract harmonic analysis. Many classical applications, in particular in the direction of L p -spaces, can be found in for example [6]. Here we extend this tool by giving a definition of a Fourier transform on the non-commutative L p -spaces associated with a locally compact quantum group. This gives a link between quantum groups and non-commutative measure theory.Recall that the Fourier transform on locally compact abelian groups can be defined in an L p -setting for p any real number between 1 and 2. This is done in the following way. Let G be a locally compact group and letĜ be its Pontrjagin dual. For a L 1 -function f on G, we define its Fourier transformf to be the function onĜ, which is defined byThenf is a continuous function onĜ vanishing at infinity. So we can consider this transform as a bounded mapthenf is a L 2 -function onĜ and this map extends to a unitary map. It is known that the Fourier transform can be generalized in a L p -setting by means of the Riesz-Thorin theorem, see [1]. The statement of this theorem directly implies the following. For any p, with 1 ≤ p ≤ 2, the linear mapf →f extends uniquely to a bounded mapThis map F p is known as the L p -Fourier transform.Date: May 10th, 2011.

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Projectivity of modules over Fourier algebras

Cited by 13 publications

References 45 publications

Multipliers, self-induced and dual Banach algebras

Multipliers, self-induced and dual Banach algebras

Homological properties of modules over semigroup algebras

The $L^p$-Fourier transform on locally compact quantum groups

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