Amenability and Invariant subspaces of the algebra of pseudomeasures

TL;DR

This paper investigates the structure of pseudomeasure algebras and their invariant subspaces to characterize the amenability of locally compact groups, establishing correspondences with subgroups and subalgebras.

Contribution

It provides new conditions for group amenability based on invariant subspaces of pseudomeasure algebras and describes bijections with subgroups and subalgebras.

Findings

Amenability characterized by invariant subspaces of $PM_\Psi(G)$

Bijection between invariant subalgebras of $PM_\Psi(G)$ and closed subgroups of $G$

Correspondence between invariant subalgebras of $A_\Phi(G)$ and compact subgroups

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 $G.$