On the Arens regularity of the Herz Algebra

TL;DR

This paper characterizes when the Herz algebra $A_p(G)$ is Arens regular, linking it to the discreteness of the group $G$ and properties of its subgroups, and explores related regularity conditions.

Contribution

It provides a complete characterization of Arens regularity for Herz algebras $A_p(G)$ based on the group's discreteness and subgroup properties, including the case of countable discrete groups.

Findings

$A_p(G)$ is Arens regular iff $G$ is discrete and all countable subgroups have Arens regular $A_p(H)$

For countable discrete $G$, relations between Arens regularity and iterated limit conditions are established

Conditions under which $l^1(G)$ as a subspace affects Arens regularity are analyzed

Abstract

Let $G$ be a locally compact group, $A_p (G)$ be the Herz algebra of $G$ associated with $1 <p< \infty$. We show that $A_p (G)$ is Arens regular if and only if $G$ is a discrete group and for each countable subgroup $H$ of $G$, $A_p (H)$ is Arens regular. In the case $G$ is a countable discrete group we investigate the relations between Arens regularity of $A_p (G)$ and the iterated limit condition. We consider the problem of Arens regularity of $l^1 (G)$ as a subspace of $A_p (G)$. A few related results when the unit ball of $(l^1 (G),.,A_p(G))$ is bounded under $\|.\|_1$-norm are also determined.