Abstract convolution function algebras over homogeneous spaces of compact groups

This paper presents a systematic study for structure of abstract Banach function ∗∗-algebras over homogeneous spaces of compact groups. Let GG be a compact group and HH be a closed subgroup of GG. Let μμ be the normalized GG-invariant measure over the homogeneous space G/HG/H associated to the Weil’s formula and 1≤p<∞1≤p<∞. Then we introduce the notions of convolution and involution for the Banach function spaces Lp(G/H,μ)Lp(G/H,μ).