GNS Construction for $$C^*$$ -Valued Positive Sesquilinear Maps on a quasi *-algebra

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