Arens regularity of the Orlicz Figà-Talamanca Herz Algebra

Abstract

Let G be a locally compact group and let $A_Φ(G)$ be the Orlicz-version of the Figà-Talamanca Herz algebra of G associated with a Young function $Φ.$ We show that if $A_Φ(G)$ is Arens regular, then $G$ is discrete. We further explore the Arens regularity of $A_Φ(G)$ when the underlying group $G$ is discrete. In the running, we also show that $A_Φ(G)$ is finite-dimensional if and only if $G$ is finite. Further, for amenable groups, we show that $A_Φ(G)$ is reflexive if and only if $G$ is finite, under the assumption that the associated Young function $Φ$ satisfies the MA-condition.

Theorems & Definitions (57)

• Lemma 3.1
• Remark 3.2
• Remark 3.3
• proof : Proof of Lemma \ref{['lemma']}
• Proposition 3.4
• proof
• Theorem 3.5
• proof
• Corollary 3.6
• proof