Measure Algebras on Homogeneous Spaces

Abstract

For a locally compact group $G$ and a compact subgroup $H$, we show that the Banach space $M(G/H)$ may be considered as a quotient space of $M(G)$. Also, we define a convolution on $M(G/H)$ which makes it into a Banach algebra. It may be identified with a closed subalgebra of the involutive Banach algebra $M(G)$, and there is no involution on $M(G/H)$ compatible with this identification unless $H$ is a normal subgroup of $G$. In other words, $M(G/H)$ is a $*$-Banach subalgebra of $M(G)$ only if $H$ is a normal subgroup of $G$. As well, it is a unital Banach algebra just when $H$ is a normal subgroup. Furthermore, when $G/H$ is attached to a strongly quasi-invariant measure, $L^1(G/H)$ is a Banach subspace of $M(G/H)$. Using the restriction of the convolution on $M(G/H)$, we obtain a Banach algebra $L^1(G/H)$, which may be considered as a Banach subalgebra of $L^1(G)$, with a right approximate identity. It has no involution and no left approximate identity except for a normal subgroup $H$. Consequently, the Banach algebra $L^1(G/H)$ is amenable if and only if $H$ is a normal subgroup and $G$ is amenable.