Convolution and involution on function spaces of homogeneous spaces

Let $G$ be a locally compact group and also let $H$ be a compact subgroup of $G$. It is shown that, if $\mu$ is a relatively invariant measure on $G\mathrm{/}H$ then there is a well-defined convolution on $L^1(G\mathrm{/}H,\mu )$ such that the Banach space $L^1(G\mathrm{/}H,\mu )$ becomes a Banach algebra. We also find a generalized definition of this convolution for other $L^p$-spaces. Finally, we show that various types of involutions can be considered on $G\mathrm{/}H$.