Banach Algebra of Complex Bounded Radon Measures on Homogeneous Space

TL;DR

This paper introduces a convolution-based Banach algebra structure on the space of bounded Radon measures on a homogeneous space, establishing its algebraic properties and relationships with related measure spaces.

Contribution

It defines a convolution on measures on homogeneous spaces and proves that this forms a non-unital Banach algebra with an approximate identity, expanding the algebraic framework for such measures.

Findings

$ M(G/H, *) $ is a non-unital Banach algebra with an approximate identity.

The algebra $ M(G/H, *) $ is not involutive.

$ L^1(G/H, *) $ is a two-sided ideal of $ M(G/H, *) $.

Abstract

Let $ H $ be a compact subgroup of a locally compact group $G$. In this paper we define a convolution on $ M(G/H) $, the space of all complex bounded Radon measures on the homogeneous space G/H. Then we prove that the measure space $ M(G/H, *) $ is a non-unital Banach algebra that possesses an approximate identity. Finally, it is shown that the Banach algebra $ M(G/H, *) $ is not involutive and also $ L^1(G/H, *) $ is a two-sided ideal of it.