Arens regularity of the Orlicz Figà-Talamanca Herz Algebra

Abstract

Let G be a locally compact group and let $A_{\Phi }(G)$ be the Orlicz-version of the Figà-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.