Amenability and invariant subspaces of the algebra of $\Psi$-pseudomeasures

Abstract

Let $G$ be a locally compact group, and let $(\Phi ,\Psi )$ be a complementary pair of Young functions satisfying the ${\Delta }_2$-condition. In this article, we consider $P{M}_{\Psi }(G)$ the Banach algebra of $\Psi$-pseudomeasures along with its predual, the Orlicz Figà-Talamanca Herz algebra $A_{\Phi }(G)$. We prove sufficient conditions for the amenability of $G$ in terms of the norm closed topologically invariant subspaces of $P{M}_{\Psi }(G)$. Further, when $G$ is amenable and the Young function $\Phi$ satisfies the MA condition, we establish a one-to-one correspondence between certain topologically invariant subalgebras of $P{M}_{\Psi }(G)$ and the closed subgroups of $G$. A similar result is obtained for $A_{\Phi }(G)$, where we derive a bijection between certain topologically invariant subalgebras of $A_{\Phi }(G)$ and the compact subgroups of $G$.