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

Abstract: For a locally compact group $G$ and a Young function $\Phi,$ let $A_\Phi(G)$ be the Orlicz analogue of the Fig`{a}-Talamanca Herz algebra $A_p(G).$ Let $\Lambda(A_\Phi(G)^{\ast\ast})$ denote the topological center of the double dual of $A_\Phi(G),$ when considered as a Banach algebra with the first Arens product. In this article, we present certain results concerning the existence of left and right identities in the Banach algebra $A_\Phi(G)^{\ast\ast}$ and explore their relationship with the space $UCB_\Psi(\widehat{G}).$ Further, for amenable groups, we establish a necessary and sufficient condition for the equality $\Lambda(A_\Phi(G)^{\ast\ast}) = A_\Phi(G)$ to hold. Moreover, we derive several results related to the semi-simplicity of the Banach algebras $A_\Phi(G)^{\ast\ast}$ and $UCB_\Psi(\widehat{G})^\ast.$