Semigroups with positive definite structure

Abstract:

Let $G$ be a semigroup with the identity 1 and an involution $x \to {x^ \ast }$. One can define a complex-valued and an ${H^ \ast }$-algebra valued positive definite function on $G$ in the obvious way. Assume that for each $x \in G$ there exists a positive number ${L_x}$ such that $q({x^ \ast }x) \leqq {L_x}q(1)$ for each complex positive definite function on $G$. It is shown that each ${H^ \ast }$-algebra valued positive definite function on $G$ is of the form $p(x) = (f,{T_x}f)$ for some representation $T$ of $G$ on a Hilbert module $H$ and $f \in H$. Also there is an analogue of Bochner theorem for $G$.