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 $P{M}_{\Psi }(G)$ and the Orlicz Figà-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 $P{M}_{\Psi }(G).$ 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 $P{M}_{\Psi }(G)$ 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 $G.$

17 pages