Covariant Function Algebras of Invariant Characters of Normal Subgroups

TL;DR

This paper develops a harmonic analysis framework for covariant function algebras associated with invariant characters of normal subgroups in locally compact groups, establishing algebraic structures and module properties.

Contribution

It introduces the structure of covariant convolution algebras and modules for invariant characters, extending harmonic analysis to this setting.

Findings

$L^1_\xi(G,N)$ forms a Banach $*$-algebra.

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

Theory applied to examples of semi-direct product groups.

Abstract

This paper presents abstract harmonic analysis foundations for structure of covariant function algebras of invariant characters of normal subgroups. Suppose that $G$ is a locally compact group and $N$ is a closed normal subgroup of $G$. Let $\xi:N\to\mathbb{T}$ be a continuous $G$-invariant character, $1\le p<\infty$, and $L_\xi^p(G,N)$ be the $L^p$-space of all covariant functions of $\xi$ on $G$. We study structure of covariant convolution in $L^p_\xi(G,N)$. It is proved that $L^1_\xi(G,N)$ is a Banach $*$-algebra and $L^p_\xi(G,N)$ is a Banach $L^1_\xi(G,N)$-module. We then investigate the theory of covariant convolutions for the case of characters of canonical normal subgroups in semi-direct product groups. The paper is concluded by realization of the theory in the case of different examples.