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A note on a question of Garth Dales: Arens regularity as a three space property

Mahmoud Filali, Jorge Galindo

TL;DR

The paper addresses whether Arens regularity is preserved under forming a three-space: an ideal and its quotient. It proves a positive reflexivity result for a class of weakly sequentially complete algebras that are direct summands of their multiplier algebras, thereby answering Garth Dales' question in that setting. It also demonstrates that Arens regularity can fail the three-space property by constructing explicit semidirect-product Banach algebras that satisfy some, but not all, needed conditions. Throughout, it develops and utilizes the notions of co-Rosenthal and Riesz ideals, and analyzes their impact on Arens regularity via quotients and $WAP$-spaces, with applications to $L^1(G)$ contexts.

Abstract

Garth Dales asked whether a Banach algebra $\mathcal{A}$ having an Arens regular closed ideal $\mathcal{J}$ with Arens regular quotient $\mathcal{A}/\mathcal{J}$ is necessarily Arens regular. We prove in this note that, for a class of Banach algebras including the standard algebras in harmonic analysis, Garth's conditions force the algebra to be even reflexive. We also give examples of Banach algebras with Garth's conditions, that are not Arens regular.

A note on a question of Garth Dales: Arens regularity as a three space property

TL;DR

The paper addresses whether Arens regularity is preserved under forming a three-space: an ideal and its quotient. It proves a positive reflexivity result for a class of weakly sequentially complete algebras that are direct summands of their multiplier algebras, thereby answering Garth Dales' question in that setting. It also demonstrates that Arens regularity can fail the three-space property by constructing explicit semidirect-product Banach algebras that satisfy some, but not all, needed conditions. Throughout, it develops and utilizes the notions of co-Rosenthal and Riesz ideals, and analyzes their impact on Arens regularity via quotients and -spaces, with applications to contexts.

Abstract

Garth Dales asked whether a Banach algebra having an Arens regular closed ideal with Arens regular quotient is necessarily Arens regular. We prove in this note that, for a class of Banach algebras including the standard algebras in harmonic analysis, Garth's conditions force the algebra to be even reflexive. We also give examples of Banach algebras with Garth's conditions, that are not Arens regular.
Paper Structure (4 sections, 10 theorems, 40 equations)

This paper contains 4 sections, 10 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Some special ideals of $L^1(G)$
  3. Riesz and co-Rosenthal ideals
  4. Examples

Key Result

Lemma 1.3

For a closed ideal ${\mathcal{J}}$ in $\mathscr{A}$, the following statements are equivalent.

Theorems & Definitions (23)

  • Definition 1.1
  • Remark 1.2
  • Lemma 1.3
  • proof
  • Corollary 1.4
  • Lemma 1.5
  • Lemma 1.6
  • Lemma 1.7
  • proof
  • Definition 1.8
  • ...and 13 more