Abstract measure algebras over homogeneous spaces of compact groups

This paper presents a systematic study for abstract Banach measure algebras over homogeneous spaces of compact groups. Let H be a closed subgroup of a 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 complex measures over G/H. Then we introduce the abstract notions of convolution and involution over the Banach measure space M(G/H).