Harmonic Analysis of Covariant Functions of Characters of Normal Subgroups

TL;DR

This paper explores the harmonic analysis of covariant functions associated with characters of normal subgroups in locally compact groups, establishing isometric isomorphisms with quotient and dual spaces of the group algebra.

Contribution

It introduces a new framework linking covariant functions to quotient spaces and dual spaces of the group algebra, extending harmonic analysis techniques.

Findings

$L^1_\xi(G,N)$ is isometrically isomorphic to a quotient of $L^1(G)$

The dual space $L^1_\xi(G,N)^*$ is isometrically isomorphic to $L^_\xi(G,N)$

Provides a harmonic analysis framework for covariant functions of characters

Abstract

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 and $L_\xi^1(G,N)$ is the $L^1$-space of all covariant functions of $\xi$ on $G$. We showed that $L^1_\xi(G,N)$ is isometrically isomorphic to a quotient space of $L^1(G)$. It is also proved that the dual space $L^1_\xi(G,N)^*$ is isometrically isomorphic to $L^\infty_\xi(G,N)$.