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 $μ$ is a relatively invariant measure on $G/H$ then there is a well-defined convolution on $L^1(G/H,μ)$ such that the Banach space $L^1(G/H,μ)$ 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 $G/H$.

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