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Multipliers, self-induced and dual Banach algebras

100%
Daws M.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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.
2
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Characterising weakly almost periodic functionals on the measure algebra

96%
Daws M.
Studia Mathematica
|
2011
|
tom 204
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nr 3
213-234
EN
Let G be a locally compact group, and consider the weakly almost periodic functionals on M(G), the measure algebra of G, denoted by WAP(M(G)). This is a C*-subalgebra of the commutative C*-algebra M(G)*, and so has character space, say $K_{WAP}$. In this paper, we investigate properties of $K_{WAP}$. We present a short proof that $K_{WAP}$ can naturally be turned into a semigroup whose product is separately continuous; at the Banach algebra level, this product is simply the natural one induced by the Arens products. This is in complete agreement with the classical situation when G is discrete. A study of how $K_{WAP}$ is related to G is made, and it is shown that $K_{WAP}$ is related to the weakly almost periodic compactification of the discretisation of G. Similar results are shown for the space of almost periodic functionals on M(G).
3
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Dual Banach algebras: representations and injectivity

96%
Daws M.
Studia Mathematica
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2007
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tom 178
|
nr 3
231-275
EN
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 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.
4
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Erratum to "Can $ℬ(ℓ^{p})$ ever be amenable?" (Studia Math. 188 (2008), 151-174)

61%
Daws M., Runde V.
Studia Mathematica
|
2009
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tom 195
|
nr 3
297-298
5
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Can $ℬ(ℓ^{p})$ ever be amenable?

61%
Daws M., Runde V.
Studia Mathematica
|
2008
|
tom 188
|
nr 2
151-174
EN
It is known that $ℬ(ℓ^{p})$ is not amenable for p = 1,2,∞, but whether or not $ℬ(ℓ^{p})$ is amenable for p ∈ (1,∞) ∖ {2} is an open problem. We show that, if $ℬ(ℓ^{p})$ is amenable for p ∈ (1,∞), then so are $ℓ^{∞}(ℬ(ℓ^{p}))$ and $ℓ^{∞}(𝓚(ℓ^{p}))$. Moreover, if $ℓ^{∞}(𝓚(ℓ^{p}))$ is amenable so is $ℓ^{∞}(𝕀,𝓚(E))$ for any index set 𝕀 and for any infinite-dimensional $ℒ^{p}$-space~E; in particular, if $ℓ^{∞}(𝓚(ℓ^{p}))$ is amenable for p ∈ (1,∞), then so is $ℓ^{∞}(𝓚(ℓ^{p} ⊕ ℓ²))$. We show that $ℓ^{∞}(𝓚(ℓ^{p} ⊕ ℓ²))$ is not amenable for p = 1,∞, but also that our methods fail us if p ∈ (1,∞). Finally, for p ∈ (1,2) and a free ultrafilter 𝒰 over ℕ, we exhibit a closed left ideal of $(𝓚(ℓ^{p}))_{𝒰}$ lacking a right approximate identity, but enjoying a certain very weak complementation property.
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