GNS Construction for \(C^*\)-Valued Positive Sesquilinear Maps on a quasi *-algebra

The GNS construction for positive invariant sesquilinear forms on quasi *-algebra \((\mathfrak A,{\mathfrak A}_{\scriptscriptstyle 0})\) $(\mathfrak{A},{\mathfrak{A}}_0)$ is generalized to a class of positive sesquilinear maps from \(\mathfrak A\times \mathfrak A\) $\mathfrak{A}\times \mathfrak{A}$ into a \(C^*\) $C^{\ast }$ -algebra \({\mathfrak {C}}\) $\mathfrak{C}$ . The result is a *-representation taking values in a space of operators acting on a certain quasi-normed \({\mathfrak {C}}\) $\mathfrak{C}$ -module.