Harmonic Analysis of Covariant Functions of Characters of Normal Subgroups

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 $ξ:N\to\mathbb{T}$ is a continuous character and $L_ξ^1(G,N)$ is the $L^1$-space of all covariant functions of $ξ$ on $G$. We showed that $L^1_ξ(G,N)$ is isometrically isomorphic to a quotient space of $L^1(G)$. It is also proved that the dual space $L^1_ξ(G,N)^*$ is isometrically isomorphic to $L^\infty_ξ(G,N)$.