Dual Banach algebras: representations and injectivity

Daws, Matthew

doi:10.4064/sm178-3-3

Cited by 43 publications

(71 citation statements)

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“…That is, we have been considering a Banach algebra A to be a dual Banach algebra if A is isometrically isomorphic to a dual space, such that the product is separately weak * -continuous. However, one can weaken this to just asking for A to be isomorphic to a dual space (sometimes this gives the same notion of weak * topology; see for example [10,Section 4]). For example, in Theorem 7.1, we can weaken ι from being an isometry to being an isomorphism onto its range.…”

Section: Henceφmentioning

confidence: 99%

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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: Henceφmentioning

confidence: 99%

“…Following [47,10], we say that a Banach algebra A which is the dual of a Banach space A * is a dual Banach algebra if multiplication in A is separately weak * -continuous. This is equivalent to the canonical image of A * in A * being an A-submodule, that is, that A is a dual A-bimodule.…”

Section: Dual Banach Algebrasmentioning

confidence: 99%

Multipliers, self-induced and dual Banach algebras

Daws¹

2010

Dissertationes Math.

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“…While this theory is well developed in the abstract setting of Banach algebras and dual Banach algebras (see [12] for example) we only need it as it applies to 1 (Z), which we now review for the reader's convenience. An element µ ∈ ∞ (Z) is weakly almost periodic if the orbit of µ under the bilateral shift is a relatively weakly compact set.…”

Section: Preduals and Semigroup Compactificationsmentioning

confidence: 99%

“…Thus, for a character Ψ on B, 12) so that Ψ · (µν) = (Ψ · µ)(Ψ · ν). Therefore, for characters Ψ 1 and Ψ 2 on B,…”

Section: (Z)mentioning

confidence: 99%

“…However, there has been little investigation of what happens in natural classes of Banach algebras; see Section 2 for further details. Motivated by Sakai's classical work on the preduals of von Neumann algebras, the first named author asked in [12] whether the weak * -topology induced by c 0 (Z) is the unique way of turning 1 (Z) into a dual Banach algebra. The results of this paper answer this question negatively: preduals on the convolution algebra 1 (Z) are far from unique.…”

Section: Introductionmentioning

confidence: 99%

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Shift invariant preduals of ℓ 1(ℤ)

Daws

Haydon²,

Schlumprecht

et al. 2012

Isr. J. Math.

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The Banach space 1 (Z) admits many non-isomorphic preduals, for example, C(K) for any compact countable space K, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on 1 (Z) weak * -continuous. This is equivalent to making the natural convolution multiplication on 1 (Z) separately weak * -continuous and so turning 1 (Z) into a dual Banach algebra. We call such preduals shift-invariant. It is known that the only shift-invariant predual arising from the standard duality between C 0 (K) (for countable locally compact K) and 1 (Z) is c 0 (Z). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak * -continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c 0 . We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of Z. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to c 0 .

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Twisted Orlicz algebras, II

Öztop

Samei

2018

Mathematische Nachrichten

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Let G be a locally compact group, let Ω ∶ × → ℂ * be a 2-cocycle, and let (Φ, Ψ) be a complementary pair of strictly increasing continuous Young functions. We continue our investigation in [14] of the algebraic properties of the Orlicz space Φ ( ) with respect to the twisted convolution ⊛ coming from Ω. We show that the twisted Orlicz algebra ( Φ ( ), ⊛ ) posses a bounded approximate identity if and only if it is unital if and only if is discrete. On the other hand, under suitable condition on Ω, ( Φ ( ), ⊛ )becomes an Arens regular, dual Banach algebra. We also look into certain cohomological properties of ( Φ ( ), ⊛ ), namely amenability and Connesamenability, and show that they rarely happen. We apply our methods to compactly generated group of polynomial growth and demonstrate that our results could be applied to variety of cases.

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Dual Banach algebras: representations and injectivity

Cited by 43 publications

References 35 publications

Multipliers, self-induced and dual Banach algebras

Multipliers, self-induced and dual Banach algebras

Shift invariant preduals of ℓ 1(ℤ)

Twisted Orlicz algebras, II

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