Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Characterizations of Asplund and Tame Functionals using Arens Products

Matan Komisarchik

TL;DR

The paper extends the Arens-regularity viewpoint from weakly almost periodic functionals to the broader classes of Asplund and tame functionals on Banach algebras. It develops a unified dictionary linking orbit regularity properties on $\mathcal A$ and $\mathcal A^{**}$ via the two Arens products $\boxdot$ and $\diamond$, including fragmentation and separability criteria, through factorization through reflexive/Asplund/Rosenthal spaces. The main applications treat $L^{1}(G)$ and its dual $L^{\infty}(G)$, giving concrete characterizations of right Asplund and right tame elements via orbits of finitely additive $\{0,1\}$-valued measures; in second-countable groups these criteria reduce to orbit-countability/discreteness in the charateristic function case. The results generalize prior work of Glasner and Megrelishvili for $\ell^{1}(\mathbb{Z})$ and provide tools for matrix coefficients and group representations to fall into Asplund or tame classes, with potential impact on topological dynamics and harmonic analysis on groups.

Abstract

We investigate the interaction between Arens products on the bidual of a Banach algebra and structural regularity properties of functionals on the algebra. Building on the classical characterization of weakly almost periodic functionals via Arens regularity, we establish new analogous criteria for Asplund and tame functionals. As an application, we specialize the theory to the group algebra $L^{1}(G)$ of a locally compact group $G$. In this setting, we derive concrete characterizations of Asplund and tame elements of $L^{\infty}(G)$ using orbits of finitely additive $\{0, 1\}$-valued measures, thus generalizing a result of Glasner and Megrelishvili for $l^{1}(\mathbb{Z})$.

Characterizations of Asplund and Tame Functionals using Arens Products

TL;DR

The paper extends the Arens-regularity viewpoint from weakly almost periodic functionals to the broader classes of Asplund and tame functionals on Banach algebras. It develops a unified dictionary linking orbit regularity properties on and via the two Arens products and , including fragmentation and separability criteria, through factorization through reflexive/Asplund/Rosenthal spaces. The main applications treat and its dual , giving concrete characterizations of right Asplund and right tame elements via orbits of finitely additive -valued measures; in second-countable groups these criteria reduce to orbit-countability/discreteness in the charateristic function case. The results generalize prior work of Glasner and Megrelishvili for and provide tools for matrix coefficients and group representations to fall into Asplund or tame classes, with potential impact on topological dynamics and harmonic analysis on groups.

Abstract

We investigate the interaction between Arens products on the bidual of a Banach algebra and structural regularity properties of functionals on the algebra. Building on the classical characterization of weakly almost periodic functionals via Arens regularity, we establish new analogous criteria for Asplund and tame functionals. As an application, we specialize the theory to the group algebra of a locally compact group . In this setting, we derive concrete characterizations of Asplund and tame elements of using orbits of finitely additive -valued measures, thus generalizing a result of Glasner and Megrelishvili for .
Paper Structure (13 sections, 29 theorems, 74 equations)

This paper contains 13 sections, 29 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Banach Algebras and Arens Products
  4. Weak-Star Continuity
  5. Classes of Functionals on Banach Algebras
  6. Weakly Almost Periodic Functionals
  7. Asplund and Tame Functionals
  8. Tame Functionals
  9. Summary
  10. Further directions and examples
  11. Operator Algebra
  12. Group Representation
  13. The Banach Algebra $L^{1}(G)$ of a Locally Compact Group

Key Result

Lemma 2.13

The orbit maps ${\mathcal{O}}_{\varphi}^{(l)}$ and ${\mathcal{O}}_{\varphi}^{(r)}$ of ${\mathcal{A}}^{**}$ on ${\mathcal{A}}^{*}$ are (weak-star, weak-star) continuous. Specifically, if $b \in {\mathcal{A}}, \varphi \in {\mathcal{A}}^{*}$ and $\{\mu_{\lambda} \}_{\lambda \in \Lambda}$ is a net in ${

Theorems & Definitions (74)

  • Definition 2.6
  • Definition 2.8
  • Remark 2.10
  • Definition 2.11
  • Lemma 2.13
  • proof
  • Corollary 2.14
  • proof
  • Lemma 2.15
  • proof
  • ...and 64 more