Completely Positive Maps: Pro-C*-algebras and Hilbert modules over Pro-C*-algebras

In this paper, we begin by presenting a construction for induced representations of Hilbert modules over pro-$C^{\ast }$-algebras for a given continuous ${}^{\ast }$-morphism between pro-$C^{\ast }$-algebras. Subsequently, we describe the structure of completely positive maps between two pro-$C^{\ast }$-algebras using Paschke’s GNS construction for CP-maps on pro-$C^{\ast }$-algebras. Furthermore, through our construction, we establish a structure theorem for a $\phi -$map between two Hilbert modules over pro-$C^{\ast }$-algebras, where $\phi$ is a continuous CP-map between pro-$C^{\ast }$-algebras. We also discuss the minimality of these representations.