In this paper, we investigate the notions of multiplicative and ternary domains for completely positive (CP) maps between pro- \(C^*\) -algebras, and establish a Schwarz-like inequality for such maps which are contractive. Along with this, we study the \(\phi \) -module domain and ternary domain for a \(\phi \) -map \(\Phi \) , where \(\Phi \) is a CP-map between two Hilbert pro- \(C^*\) -modules. Through a detailed construction, we demonstrate that the ternary domain of a \(\phi \) -map \(\Phi \) coincides with the \(\phi \) -module domain of \(\Phi \) . Furthermore, we establish relationships between the multiplicative and ternary domains of a CP-map and the associated Stinespring triple. In addition, we derive connections between the Stinespring-like representation for \(\phi \) -maps and the \(\phi \) -module domain of such maps.
