Amenability and Invariant subspaces of the algebra of pseudomeasures

Abstract

Let $G$ be a locally compact group and $(\Phi,\Psi)$ a complimentary pair of Young functions. In this article, we consider the Banach algebra of $\Psi$-pseudomeasures $PM_\Psi(G)$ and the Orlicz Fig\`{a}-Talamanca Herz algebra $A_\Phi(G).$ We prove sufficient conditions for a group $G$ to be amenable in terms of the norm closed topologically invariant subspaces of $PM_\Psi(G).$ Further, for an amenable group $G$ with the Young function $\Phi$ satisfying the MA condition, we establish a one-to-one correspondence between certain topologically invariant subalgebras of $PM_\Psi(G)$ and the class of closed subgroups of $G.$ Moreover, we prove a similar result for the predual $A_\Phi(G)$ and derive a bijection between certain topologically invariant subalgebras of $A_\Phi(G)$ and the set of compact subgroups of $G.$