Ideals in the dual of introverted subspaces of $Ψ$-pseudomeasures

Abstract

Let $G$ be a locally compact group and $(Φ,Ψ)$ a complementary pair of Young functions satisfying the $Δ_2$-condition. Let $A_Φ(G)$ be the Orlicz analogue of the Figà-Talamanca Herz algebra $A_p(G).$ The dual of the algebra $A_Φ(G)$ is the space of $Ψ$-pseudomeasures, denoted by $PM_Ψ(G).$ For certain topologically introverted subspaces $\mathcal{A}$ of $PM_Ψ(G)$ and the Banach algebras $W_Φ(G)$ or $B_Φ(G),$ denoted by $\mathcal{B},$ we characterise the maximal regular left/right/two-sided ideals of the Banach algebras $\mathcal{A}^{'}$ and $\mathcal{B}^{''}$ considered with the Arens product. We further characterise the minimal left ideals of $\mathcal{A}^{'}$ and prove the necessary and sufficient conditions for the existence of minimal ideals in the algebras $A_Φ(G)$ and $\mathcal{B}.$