Generalized positive linear functionals on a Banach algebra with an involution

Abstract:

Let $A$ be a proper ${H^\ast }$-algebra and let $B$ be a Banach $\ast$-algebra with an identity. A linear mapping $\varphi :B \to A$ is called a positive $A$-functional if ${\Sigma _{i,j}}a_i^\ast \varphi (x_i^\ast {x_j}){a_j}$ is positive for all ${x_1},{x_2}, \cdots ,{x_n} \in B$ and ${a_1},{a_2}, \cdots ,{a_n} \in A$. It is shown that for each positive $A$-functional $\varphi$ there exists a $\ast$-representation $T$ of $B$ by $A$-linear operators on a Hilbert module $H$ such that $\varphi (x) = ({f_0},Tx{f_0})$ for all $x \in B$ and some ${f_0} \in H$. If $B$ is of the form $B = \{ \lambda e + x|\lambda$ complex, $e$ is the (abstract) identity, $x \in {L^1}(G)\}$ for some locally compact group $G$ then $\varphi$ has the form $\varphi (\lambda e + x) = \lambda \varphi (e) + \int {{\text { }}_G}x(t)p(t)dt$ for some generalized ($A$-valued) positive definite function $p$ on $G,x \in {L^1}(G)$.