Topological Center of the Double Dual of the Orlicz Figà-Talamanca Herz Algebra

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