Fourier–Stieltjes transforms over homogeneous spaces of compact groups

Abstract

This paper presents a unified operator theory approach to the abstract notion of Fourier–Stieltjes transforms for Banach measure algebras over homogeneous spaces of compact groups. Let $H$ be a closed subgroup of the compact group $G$ and $G/H$ be the left coset space associated to the subgroup $H$ in $G$. Also, let $M(G/H)$ be the Banach measure space consists of all (bounded) complex Radon measures over the compact homogeneous space $G/H$. We then study theoretical aspects of operator-valued Fourier–Stieltjes transform for the Banach measure algebras $M(G/H)$. We shall also present a uniqueness theorem for the abstract Fourier–Stieltjes transforms.