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

TL;DR

This paper characterizes the topological center of the double dual of the Orlicz Figà-Talamanca Herz algebra for locally compact groups, linking it to algebraic and topological properties.

Contribution

It establishes a necessary and sufficient condition for the topological center of the double dual to equal the algebra itself, extending known results to Orlicz algebra contexts.

Findings

Characterization of the topological center of the double dual of $A_\Phi(G)$

Conditions for semi-simplicity of related Banach algebras

Results on the structure of $A_\Phi(G)^{**}$ and $UCB_\Psi(\widehat{G})^*$

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 $UCB_\Psi(\widehat{G})^\ast.$