Arens regularity of the Orlicz Fig\`a-Talamanca Herz Algebra

Abstract

Let G be a locally compact group and let $A_\Phi(G)$ be the Orlicz-version of the Fig\`{a}-Talamanca Herz algebra of G associated with a Young function $\Phi.$ We show that if $A_\Phi(G)$ is Arens regular, then $G$ is discrete. We further explore the Arens regularity of $A_\Phi(G)$ when the underlying group $G$ is discrete. In the running, we also show that $A_\Phi(G)$ is finite-dimensional if and only if $G$ is finite. Further, for amenable groups, we show that $A_\Phi(G)$ is reflexive if and only if $G$ is finite, under the assumption that the associated Young function $\Phi$ satisfies the MA-condition.