$L_p^T-$ functions on locally compact groups

Abstract

Let $G$ be a locally compact group and $1\le p<\infty$. Based on some important earlier works, in this paper the concept of $L_p^T-$function is introduced. Then the structure of the space $L_p^T(G)$, which is consisting of all $L_p^T-$functions, is investigated. As an important result, $L_p^T(G{)}^{+}$ is completely characterized, for the class of amenable locally compact groups. Furthermore, it is verified some applications of Plancherel Theorem on $L_2^T(G)$, in the class of abelian locally compact groups.