Topological Center of the Double Dual of the Orlicz Fig\`{a}-Talamanca Herz Algebra

Abstract

Let $G$ a locally compact group and $(\Phi,\Psi)$ be a complementary pair of Young functions. Let $A_\Phi(G)$ be the Orlicz analogue of the classical Fig\`{a}-Talamanca Herz algebra $A_p(G).$ In this article, we establish a necessary and sufficient condition for the equality $\Lambda(A_\Phi(G)^{\ast\ast}) = A_\Phi(G)$ to hold, where $\Lambda(A_\Phi(G)^{\ast\ast})$ denotes the topological center of the double dual of $A_\Phi(G)$ when equipped with the first Arens product. Furthermore, we prove several results concerning the semi-simplicity of the Banach algebras $A_\Phi(G)^{\ast\ast}$ and $UCB_\Psi(\widehat{G})^\ast.$