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

Abstract

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