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Radicals of Biduals of Beurling Algebras Can Be Different for the Two Arens Products

Jared T. White

TL;DR

The paper resolves a question of Dales and Lau by constructing a Beurling algebra on the free group $ ext{F}_3$ with a non-symmetric weight $\omega$ from an infinite generating set $X$, for which the Jacobson radical of the bidual differs between the two Arens products $oxed{oxempty}$ and $oxed{oxempty}$. The method builds $ ext{A}= ext{l}^1( ext{F}_3,\omega)$ and introduces weak*-limit functionals $\Phi_0$ and $\\Psi_0$ so that $\, ext{I}= ext{A}^{**} oxed{ ext{rad}}$ under $oxempty$ contains a nonzero element with nilpotent left action, while the analogous right action under $ riangle$ is not radical; a second functional $\, extPsi_0$ ensures non-quasi-nilpotence of $\, extPhi_0 riangle extPsi_0$. The result demonstrates that $ ext{rad}( ext{A}^{**}, oxempty) e ext{rad}( ext{A}^{**}, riangle)$ and provides a nontrivial example beyond the commutative/operator-algebra cases, enriching the understanding of bidual radicals and Arens regularity.

Abstract

Let $\operatorname{rad}$ denote the Jacobson radical of a Banach algebra, and let $\Box$ and $\Diamond$ denote the two Arens products on its bidual. We give an example of a Beurling algebra $\mathcal{A}$ for which $\operatorname{rad}(\mathcal{A}^{**}, \Box) \neq \operatorname{rad}(\mathcal{A}^{**}, \Diamond)$, answering a question of Dales and Lau. The underlying group in our example is the free group on three generators.

Radicals of Biduals of Beurling Algebras Can Be Different for the Two Arens Products

TL;DR

The paper resolves a question of Dales and Lau by constructing a Beurling algebra on the free group with a non-symmetric weight from an infinite generating set , for which the Jacobson radical of the bidual differs between the two Arens products and . The method builds and introduces weak*-limit functionals and so that under contains a nonzero element with nilpotent left action, while the analogous right action under is not radical; a second functional ensures non-quasi-nilpotence of . The result demonstrates that and provides a nontrivial example beyond the commutative/operator-algebra cases, enriching the understanding of bidual radicals and Arens regularity.

Abstract

Let denote the Jacobson radical of a Banach algebra, and let and denote the two Arens products on its bidual. We give an example of a Beurling algebra for which , answering a question of Dales and Lau. The underlying group in our example is the free group on three generators.
Paper Structure (10 sections, 18 theorems, 108 equations)

This paper contains 10 sections, 18 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. Groups and Basic Notation
  4. Banach Algebras
  5. Nets and Subnets
  6. The Stone--Čech Compactification
  7. Outline of the Proof
  8. Basic Lemmas For Calculating Word Lengths
  9. Proof of the Key Word Length Results
  10. Proof of the Main Result

Key Result

Lemma 3.1

Let $\mathcal{A}$ be a commutative Banach algebra. Then ${\rm rad\,}(\mathcal{A}^{**}, \Box) = {\rm rad\,}(\mathcal{A}^{**}, \Diamond)$.

Theorems & Definitions (37)

  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Corollary 4.4
  • proof
  • ...and 27 more