Lebesgue type decomposition of subspaces of Fourier-Stieltjes algebras

Kaniuth, Eberhard; Lau, Anthony To-Ming; Schlichting, Günter

doi:10.1090/s0002-9947-02-03203-8

Cited by 9 publications

(4 citation statements)

References 38 publications

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“…As already remarked, eA * * is isometrically isomorphic to the multiplier algebra M (A). The Lebesgue decomposition theorem and its extension to the Fourier-Stieltjes algebra, see [14] and [13], show that group algebras on compact groups and Fourier algebras on discrete amenable groups are both standard wscai algebras. Theorem 1.9.…”

Section: Introductionmentioning

confidence: 99%

Extreme non-Arens regularity of the group algebra

Filali

Galindo

2018

Forum Mathematicum

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As is well-know, on an Arens regular Banach algebra all continuous functionals are weakly almost periodic. In this paper we show that ℓ 1 -bases which approximate upper and lower triangles of products of elements in the algebra produce large sets of functionals that are not weakly almost periodic. This leads to criteria for extreme non-Arens regularity of Banach algebras in the sense of Granirer. We find in particular that bounded approximate identities (bai's) and bounded nets converging to invariance (TI-nets) both fall into this approach, suggesting that this is indeed the main tool behind most known constructions of non-Arens regular algebras.These criteria can be applied to the main algebras in harmonic analysis such as the group algebra, the measure algebra, the semigroup algebra (with certain weights) and the Fourier algebra. In this paper, we apply our criteria to the Lebesgue-Fourier algebra, the 1-Segal Fourier algebra and the Figà-Talamanca

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Section: Introductionmentioning

confidence: 99%

Extreme non-Arens regularity of the group algebra

Filali

Galindo

2018

Forum Mathematicum

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“…Our assumption implies that there is a continuous unitary representation σ of G that is disjoint from λ G such that π = λ G ⊕σ and A = A(G)⊕ 1 A s (G), where A s (G) = A σ is what we shall call the singular part of A [1, Proposition 3.12, Corollary 3.13 and Theorem 3.18]. When A = B(G), this is the Lebesgue decomposition of B(G) that was studied in [25] and [22]. With the following lemmas, we record more useful facts for future reference.…”

Section: Preliminary Resultsmentioning

confidence: 99%

Homomorphisms of Fourier–Stieltjes algebras

Stokke¹

2021

Studia Math.

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Every homomorphism ϕ : B(G) → B(H) between Fourier-Stieltjes algebras on locally compact groups G and H is determined by a continuous mapping α : Y → ∆(B(G)), where Y is a set in the open coset ring of H and ∆(B(G)) is the Gelfand spectrum of B(G) (a *-semigroup). We exhibit a large collection of maps α for which ϕ = jα : B(G) → B(H) is a completely positive/completely contractive/completely bounded homomorphism and establish converse statements in several instances. For example, we fully characterize all completely positive/completely contractive/completely bounded homomorphisms ϕ : B(G) → B(H) when G is a Euclidean or p-adic motion group. In these cases, our description of the completely positive/completely contractive homomorphisms employs the notion of a "fusion map of a compatible system of homomorphisms/affine maps" and is quite different from the Fourier algebra situation.

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“…We begin with a few preliminaries, starting with the following Lemma that basically follows from [22,Lemma 1.1]. Recall (see [30] or [42]) that B(Γ) = A(Γ) ⊕ B s (Γ) where B s (Γ) is a closed translation invariant subspace of B(Γ). If φ = φ a + φ s ∈ B(Γ), then φ = φ a + φ s .…”

Section: Is Reflexive If and Only If It Has Finite Dimensionmentioning

confidence: 99%

Arens regularity of ideals of the group algebra of a compact Abelian group

Esmailvandi,

Filali,

Galindo

2023

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Let $G$ be a compact Abelian group and $E$ a subset of the group $\widehat {G}$ of continuous characters of $G$ . We study Arens regularity-related properties of the ideals $L_E^1(G)$ of $L^1(G)$ that are made of functions whose Fourier transform is supported on $E\subseteq \widehat {G}$ . Arens regularity of $L_E^1(G)$ , the centre of $L_E^1(G)^{\ast \ast }$ and the size of $L_E^1(G)^\ast /\mathcal {WAP}(L_E^1(G))$ are studied. We establish general conditions for the regularity of $L_E^1(G)$ and deduce from them that $L_E^1(G)$ is not strongly Arens irregular if $E$ is a small-2 set (i.e. $\mu \ast \mu \in L^1(G)$ for every $\mu \in M_E^1(G)$ ), which is not a $\Lambda (1)$ -set, and it is extremely non-Arens regular if $E$ is not a small-2 set. We deduce also that $L_E^1(G)$ is not Arens regular when $\widehat {G}\setminus E$ is a Lust-Piquard set.

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Lebesgue type decomposition of subspaces of Fourier-Stieltjes algebras

Cited by 9 publications

References 38 publications

Extreme non-Arens regularity of the group algebra

Extreme non-Arens regularity of the group algebra

Homomorphisms of Fourier–Stieltjes algebras

Arens regularity of ideals of the group algebra of a compact Abelian group

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