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

TL;DR

This paper investigates the conditions under which a Banach algebra is Arens regular, showing that certain conditions imply reflexivity and providing examples that challenge previous assumptions about Arens regularity.

Contribution

It proves that for a class of Banach algebras, Garth Dales' conditions imply the algebra is reflexive, and it provides counterexamples to the conjecture that these conditions ensure Arens regularity.

Findings

Garth Dales' conditions imply reflexivity in certain Banach algebras.

Counterexamples exist where the conditions do not guarantee Arens regularity.

The results include standard algebras in harmonic analysis.

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.