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

This paper presents a systematic theoretical study for the abstract notion of operator-valued Fourier transforms over homogeneous spaces of compact groups. Let GG be a compact group, HH be a closed subgroup of GG, and μμ be the normalized GG-invariant measure over the left coset space G/HG/H associated to the Weil's formula. We introduce the generalized notions of abstract dual homogeneous space G/HˆG/H^ for the compact homogeneous space G/HG/H and also the operator-valued Fourier transform over the Banach function space L1(G/H,μ)L1(G/H,μ). We prove that the abstract Fourier transform over G/HG/H satisfies the Plancherel formula and the Poisson summation formula.