scholarly journals APPROXIMATE AMENABILITY OF SEGAL ALGEBRAS

2013 ◽  
Vol 95 (1) ◽  
pp. 20-35 ◽  
Author(s):  
MAHMOOD ALAGHMANDAN
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Amenable Group ◽  
Locally Compact Abelian Group ◽  
Compact Groups ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Segal Algebra ◽  
Approximate Amenability ◽  
Segal Algebras

AbstractIn this paper we first show that for a locally compact amenable group $G$, every proper abstract Segal algebra of the Fourier algebra on $G$ is not approximately amenable; consequently, every proper Segal algebra on a locally compact abelian group is not approximately amenable. Then using the hypergroup generated by the dual of a compact group, it is shown that all proper Segal algebras of a class of compact groups including the $2\times 2$ special unitary group, $\mathrm{SU} (2)$, are not approximately amenable.

scholarly journals Constructions of the maximal strongly character invariant segal algebras and their applications

1983 ◽  
Vol 35 (1) ◽  
pp. 123-131
Author(s):  
Chang-Pao Chen
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Banach Algebras ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Segal Algebra ◽  
Segal Algebras

AbstractLet G denote any locally compact abelian group with the dual group Γ. We construct a new kind of subalgebra L1(G) ⊗ΓS of L1(G) from given Banach ideal S of L1(G). We show that L1(G) ⊗гS is the larger amoung all strongly character invariant homogeneous Banach algebras in S. when S contains a strongly character invariant Segal algebra on G, it is show that L1(G) ⊗гS is also the largest among all strongly character invariant Segal algebras in S. We give applications to characterizations of two kinds of subalgebras of L1(G)-strongly character invariant Segal algebras on G and Banach ideal in L1(G) which contain a strongly character invariant Segal algebra on G.

scholarly journals On weak spectral synthesis

1991 ◽  
Vol 43 (2) ◽  
pp. 279-282 ◽  
Author(s):  
K. Parthasarathy ◽  
Sujatha Varma
Keyword(s):  
Abelian Group ◽  
Group Algebra ◽  
Lie Group ◽  
Compact Abelian Group ◽  
Spectral Synthesis ◽  
Locally Compact Abelian Group ◽  
Fourier Algebra ◽  
Compact Lie Group ◽  
Locally Compact

Weak spectral synthesis fails in the group algebra and the generalised group algebra of any non compact locally compact abelian group and also in the Fourier algebra of any infinite compact Lie group.

scholarly journals The algebra of functions with Fourier transforms in a given function space

1973 ◽  
Vol 9 (1) ◽  
pp. 73-82 ◽  
Author(s):  
U.B. Tewari ◽  
A.K. Gupta
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Function Space ◽  
Compact Abelian Group ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Algebra Of Functions

Let G be a locally compact abelian group and Ĝ be its dual group. For 1 ≤ p < ∞, let Ap (G) denote the set of all those functions in L1(G) whose Fourier transforms belong to Lp (Ĝ). Let M(Ap (G)) denote the set of all functions φ belonging to L∞(Ĝ) such that is Fourier transform of an L1-function on G whenever f belongs to Ap (G). For 1 ≤ p < q < ∞, we prove that Ap (G) Aq(G) provided G is nondiscrete. As an application of this result we prove that if G is an infinite compact abelian group and 1 ≤ p ≤ 4 then lp (Ĝ) M(Ap(G)), and if p > 4 then there exists ψ є lp (Ĝ) such that ψ does not belong to M(Ap (G)).

scholarly journals A singular convolution kernel without pseudo-periods

1981 ◽  
Vol 83 ◽  
pp. 1-4
Author(s):  
Jesper Laub
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Convolution Kernel ◽  
Locally Compact ◽  
Excessive Measures

Let G be a locally compact abelian group and N a non-zero convolution kernel on G satisfying the domination principle. We define the cone of N-excessive measures E(N) to be the set of positive measures ξ for which N satisfies the relative domination principle with respect to ξ. For ξ ∈ E(N) and Ω ⊆ G open the reduced measure of ξ over Ω is defined as.

scholarly journals Pure point/continuous decomposition of translation-bounded measures and diffraction

2018 ◽  
Vol 40 (2) ◽  
pp. 309-352
Author(s):  
JEAN-BAPTISTE AUJOGUE
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Diffraction Theory ◽  
Natural Generalization ◽  
Pure Point ◽  
Locally Compact Abelian Group ◽  
Uniqueness Property ◽  
Locally Compact ◽  
Continuous Decomposition

In this work we consider translation-bounded measures over a locally compact Abelian group$\mathbb{G}$, with a particular interest in their so-called diffraction. Given such a measure$\unicode[STIX]{x1D714}$, its diffraction$\widehat{\unicode[STIX]{x1D6FE}}$is another measure on the Pontryagin dual$\widehat{\mathbb{G}}$, whose decomposition into the sum$\widehat{\unicode[STIX]{x1D6FE}}=\widehat{\unicode[STIX]{x1D6FE}}_{\text{p}}+\widehat{\unicode[STIX]{x1D6FE}}_{\text{c}}$of its atomic and continuous parts is central in diffraction theory. The problem we address here is whether the above decomposition of$\widehat{\unicode[STIX]{x1D6FE}}$lifts to$\unicode[STIX]{x1D714}$itself, that is to say, whether there exists a decomposition$\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{p}}+\unicode[STIX]{x1D714}_{\text{c}}$, where$\unicode[STIX]{x1D714}_{\text{p}}$and$\unicode[STIX]{x1D714}_{\text{c}}$are translation-bounded measures having diffraction$\widehat{\unicode[STIX]{x1D6FE}}_{\text{p}}$and$\widehat{\unicode[STIX]{x1D6FE}}_{\text{c}}$, respectively. Our main result here is the almost sure existence, in a sense to be made precise, of such a decomposition. It will also be proved that a certain uniqueness property holds for the above decomposition. Next, we will be interested in the situation where translation-bounded measures are weighted Meyer sets. In this context, it will be shown that the decomposition, whether it exists, also consists of weighted Meyer sets. We complete this work by discussing a natural generalization of the considered problem.

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