CONVOLUTION AND HOMOGENEOUS SPACES

Let G be a locally compact Hausdorff topological groupaud H be a compact subgroup of G. Then, the homogeneous spaceG/ H possesses a specific Radon measure, which is called a relativelyinvariant measure. We show that the concepts of convolutionand involution can be extended to the integrable functions definedon this homogeneous space. We study the properties of convolutionand prove that the space of integrable functions is an involutiveBanach algebra with an approximate identity. We also find a necessaryand sufficient condition on a closed subspace of this Banachalgebra to make it a left ideal