Abstract convolution function algebras over homogeneous spaces of compact groups

Abstract

This paper presents a systematic study for structure of abstract Banach function $*$-algebras over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $\mu$ be the normalized $G$-invariant measure over the homogeneous space $G/H$ associated to the Weil’s formula and $1\leq p<\infty$. Then we introduce the notions of convolution and involution for the Banach function spaces $L^{p}(G/H,\mu)$.