Abstract operator-valued Fourier transforms over homogeneous spaces of compact groups (Q1691968)

Summary: 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.