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