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 Figà-Talamanca Herz algebra $A_p(G).$ The dual of the algebra $A_{\Phi }(G)$ is the space of $\Psi$-pseudomeasures, denoted by $P{M}_{\Psi }(G).$ For certain topologically introverted subspaces $\mathcal{A}$ of $P{M}_{\Psi }(G)$ 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 ${\mathcal{A}}^{{}^{\prime }}$ and ${\mathcal{B}}^{{}^{\prime \prime }}$ considered with the Arens product. We further characterise the minimal left ideals of ${\mathcal{A}}^{{}^{\prime }}$ and prove the necessary and sufficient conditions for the existence of minimal ideals in the algebras $A_{\Phi }(G)$ and $\mathcal{B}.$

15 pages