On the Arens regularity of the Herz Algebra

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 $\Vert .{\Vert }_1$-norm are also determined.