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On the BSE- property of vector valued Beurling algebra $L^1(G,ω, \mathcal{A})$

Jekwin Dabhi, Prakash Dabhi

Abstract

Let $G$ be a locally compact abelian group, and let $ω:G \to [1,\infty)$ be a measurable weight, i.e., $ω$ is measurable, and $ω(s+t)\leq ω(s)ω(t)$ for all $s, t \in G$. Let $\mathcal{A}$ be a semisimple commutative Banach algebra with a predual $\mathcal A_\ast$ such that the Gel'fand space $Φ_{\mathcal A}\subset \mathcal{A}_\ast$. If $ω^{-1}$ is vanishing at infinity, then we show that the Banach algebra $L^1(G,ω,\mathcal{A})$ is a BSE- algebra if and only if $\mathcal A$ is a BSE- algebra.

On the BSE- property of vector valued Beurling algebra $L^1(G,ω, \mathcal{A})$

Abstract

Let be a locally compact abelian group, and let be a measurable weight, i.e., is measurable, and for all . Let be a semisimple commutative Banach algebra with a predual such that the Gel'fand space . If is vanishing at infinity, then we show that the Banach algebra is a BSE- algebra if and only if is a BSE- algebra.
Paper Structure (4 sections, 9 theorems, 40 equations)

This paper contains 4 sections, 9 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. $L^1(G,\mathcal{A})$ is a BSE- algebra.
  3. $L^1(G,\omega, \mathcal{A})$ is BSE
  4. Examples

Key Result

Theorem 1.1

Let $\mathcal{A}$ be a semisimple commutative Banach algebra, let $\mathcal{A}$ have a predual $\mathcal{A}_\ast$ such that $\Phi_{\mathcal{A}} \subset \mathcal{A}_\ast$, and let $G$ be a locally compact abelian group. Then $L^1(G, \mathcal{A})$ is a BSE-algebra if and only if $\mathcal{A}$ is a BSE

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['B1']}
  • Lemma 3.1
  • proof
  • ...and 8 more