Topological Center of the Double Dual of the Orlicz Figà-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 Figà-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 $UC{B}_{\Psi }(\hat{G}{)}^{\ast }.$

17 pages