The Representations and Positive Type Functions of Some Homogenous Spaces

ýFor a homogeneous spaces ý$ýG/Hý$ý, 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 ý$ýGý$ý. ýAlso we prove that there exists a one to one correspondence between nondegenerat $ast$-representations of $L^1(G/H)$ and representations of $G/H$ý. We propose a relation between cyclic representations of $L^1(G/H)$ and positive type functions on $G/H$ý. We prove that the Gelfand Raikov theorem for $G/H$ holds if and only if $H$ is normalý.