Approximate Identities for Ideals of Segal Algebras on A Compact Group

“…On the other hand, it has been shown in [19] that for a compact group G every closed two-sided ideal of L 1 (G) has an approximate identity satisfying condition (U). As a consequence, It is inspiring to recall here the so-called Separable Extension Problem.…”

Section: Proposition 21 Suppose That X Is a Banach Space Having Thementioning

confidence: 99%

“…Here⊗ denotes projective tensor product. Further investigation made in [19] suggests that the geometric property described in this notion may play an important role in studying ideals of Banach algebras.…”

Section: Introductionmentioning

confidence: 99%

“…Many examples reveal that, in dealing with ideals of Banach algebras, more often than not the existence of an approximate identity that satisfies condition (U) can be asserted in terms of approximate complementation of the ideal. For instance, in a Segal algebra on a compact group, a closed right or left ideal has a left or right approximate identity satisfying condition (U) if and only if it is approximately complemented as a subspace in the Segal algebra [19].…”

Section: Introductionmentioning

confidence: 99%

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Approximate complementation and its applications in studying ideals of Banach algebras

Zhang¹

2003

MATH. SCAND.

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We show that a subspace of a Banach space having the approximation property inherits this property if and only if it is approximately complemented in the space. For an amenable Banach algebra a closed left, right or two-sided ideal admits a bounded right, left or two-sided approximate identity if and only if it is bounded approximately complemented in the algebra. If an amenable Banach algebra has a symmetric diagonal, then a closed left (right) ideal J has a right (resp. left) approximate identity (p α ) such that, for every compact subset K of J , the net (a · p α ) (resp. (p α · a)) converges to a uniformly for a ∈ K if and only if J is approximately complemented in the algebra.

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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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Approximate Identities for Ideals of Segal Algebras on A Compact Group

Cited by 6 publications

References 14 publications

Biflatness and Pseudo-Amenability of Segal Algebras

Biflatness and Pseudo-Amenability of Segal Algebras

Approximate complementation and its applications in studying ideals of Banach algebras

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

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