Ideals in the dual of introverted subspaces of $\Psi$-pseudomeasures

Abstract

Let $G$ be a locally compact group and $(\Phi,\Psi)$ a complementary pair of Young functions satisfying the $\Delta_2$-condition. Let $A_\Phi(G)$ be the Orlicz analogue of the Fig\`{a}-Talamanca Herz algebra $A_p(G).$ The dual of the algebra $A_\Phi(G)$ is the space of $\Psi$-pseudomeasures, denoted by $PM_\Psi(G).$ For certain topologically introverted subspaces $\mathcal{A}$ of $PM_\Psi(G)$ and the Banach algebras $W_\Phi(G)$ or $B_\Phi(G),$ 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 $A_\Phi(G)$ and $\mathcal{B}.$