Arens regularity of \(A_\Phi (G)\)

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