Covariant Function Algebras of Invariant Characters of Normal Subgroups

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.