The Representations and Positive Type Functions of Some Homogenous Spaces

‎For a homogeneous spaces ‎$‎G/H‎$‎, we show that the convolution on $L^1(G/H)$ is the same as convolution on $L^1(K)$, where $G$ is semidirect product of a closed subgroup $H$ and a normal subgroup $K $ of ‎$‎G‎$‎. ‎Also we prove that there exists a one to one correspondence between nondegenerat $ast$-representations of $L^1(G/H)$ and representations of $G/H$‎. We propose a relation between cyclic representations of $L^1(G/H)$ and positive type functions on $G/H$‎. We prove that the Gelfand Raikov theorem for $G/H$ holds if and only if $H$ is normal‎.