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.

Type of Study: Research paper |
Subject:
General