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 $\xi :N\to \mathbb{T}$ 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 $L_{\xi }^1(G,N)$ is isometrically isomorphic to a quotient space of $L^1(G)$. It is also proved that the dual space $L_{\xi }^1(G,N{)}^{\ast }$ is isometrically isomorphic to $L_{\xi }^{\infty }(G,N)$.

Corrected typos, slight revisions, and examples removed. arXiv admin note: text overlap with arXiv:2102.07892