Abstract operator-valued Fourier transforms over homogeneous spaces of compact groups

Abstract

This paper presents a systematic theoretical study for the abstract notion of operator-valued Fourier transforms over homogeneous spaces of compact groups. Let $G$ be a compact group, $H$ be a closed subgroup of $G$, and $\mu$ be the normalized $G$-invariant measure over the left coset space $G/H$ associated to the Weil's formula. We introduce the generalized notions of abstract dual homogeneous space $\widehat{G/H}$ for the compact homogeneous space $G/H$ and also the operator-valued Fourier transform over the Banach function space $L^1(G/H,\mu)$. We prove that the abstract Fourier transform over $G/H$ satisfies the Plancherel formula and the Poisson summation formula.