Banach Convolution Modules of Group Algebras on Covariant Functions of Characters of Normal Subgroups

TL;DR

This paper explores the structure of Banach convolution modules formed by group algebras acting on covariant functions associated with characters of normal subgroups in locally compact groups, with a focus on semi-direct products.

Contribution

It establishes that these covariant function spaces are Banach modules over the group algebra and analyzes their convolution module actions in specific group settings.

Findings

L^p_\xi(G,N) is a Banach L^1(G)-module.

Characterizes convolution actions on covariant functions.

Provides structure results for semi-direct product groups.

Abstract

This paper investigates structure of Banach convolution modules induced by group algebras on covariant functions of characters of closed normal subgroups. Let $G$ be a locally compact group with the group algebra $L^1(G)$ and $N$ be a closed normal subgroup of $G$. Suppose that $\xi:N\to\mathbb{T}$ is a continuous character, $1\le p<\infty$ and $L_\xi^p(G,N)$ is the $L^p$-space of all covariant functions of $\xi$ on $G$. It is shown that $L^p_\xi(G,N)$ is a Banach $L^1(G)$-module. We then study convolution module actions of group algebras on covariant functions of characters for the case of canonical normal subgroups in semi-direct product groups.